ON THE INTEGRALITY OF THE TAYLOR COEFFICIENTS OF MIRROR MAPS

C. KRATTENTHALER^{†} AND T. RIVOAL

Abstract. We show that the Taylor coefficients of the series q(z) = z exp(G(z)/F(z)) are integers, where F(z) and G(z) + log(z)F(z) are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z = 0. We also address the question of finding the largest integer u such that the Taylor coefficients of (z^{−1}q(z))^{1/u} are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi–Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general “integrality” conjecture of Zudilin about these mirror maps. A further outcome of the present study is the determination of the Dwork–Kontsevich sequence (u_{N})_{N≥1}, where u_{N} is the largest integer such that q(z)^{1/u_{N}} is a series with integer coefficients, where q(z) = exp(F (z)/G(z)), F (z) = ∑_{m=0}^{∞}(Nm)! z^{m}/m!^{N} and G(z) = ∑_{m=1}^{∞}(H_{Nm} − H_{m})(Nm)! z^{m}/m!^{N}, with H_{n} denoting the n-th harmonic number, conditional on the conjecture that there are no prime number p and integer N such that the p-adic valuation of H_{N} −1 is strictly greater than 3.

1. Introduction and statement of results

1.1. Mirror maps. Mirror maps have appeared quite recently in mathematics and physics. Indeed, the term “mirror map” was coined in the late 1980s by physicists whose research in string theory led them to discover deep facts in algebraic geometry (e.g., pair of Calabi–Yau manifolds in “mirror,” enumeration of rational curves on one of these Calabi– Yau manifolds in dimension 3). In a sense, mirror maps can be viewed as higher dimensional generalisations of certain classical modular forms, which naturally appear in low dimensions (see some examples below).

The purpose of the present article is to prove rather sharp integrality assertions for the Taylor coefficients of certain mirror maps coming from hypergeometric differential equations, which are Picard–Fuchs equations of suitable one parameter families of Calabi– Yau complete intersections in weighted projective spaces. The corresponding results (see Theorems 1–5) encompass integrality results on these mirror maps which exist in the literature, improving and refining them in numerous cases.

Date: June 16, 2008. 2000 Mathematics Subject Classification. Primary 11S80; Secondary 11J99 14J32 33C20. Key words and phrases. Calabi–Yau manifolds, integrality of mirror maps, p-adic analysis, Dwork’s

theory, harmonic numbers, hypergeometric differential equations.

†Research partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

Before getting deeper into the subject, it is beneficial to define at this point a special case of a mirror map (see just after (1.1) below), which will be studied in great detail in the present paper. For a real number x > 0 and an integer m ≥ 0, set

H(x, m) := ∑_{n=0}^{m−1} \frac{1}{x + n},

where, by convention, the empty sum is 0. When x = 1, H(1, m) is simply the m-th harmonic number H_{m}. For integers k ≥ 1 and N ≥ 1, let us define the power series

F_{N}(z) := ∑_{m=0}^{∞} \frac{(Nm)!^{k}}{m!^{kN}} z^{m}

and

G_{N}(z) := k ∑_{m=1}^{∞} \frac{(Nm)!^{k}}{m!^{kN}} ( ∑_{j=1}^{N−1} H(j/N, m) − (N − 1)H(1, m)) z^{m},

which converge for |z| < 1/N^{kN}. (^{1}) The functions F_{N}(z) and G_{N}(z) + log(z)F_{N}(z) are solutions of the same hypergeometric differential equation, which is a special case of (1.18) below. (It is of maximal unipotent monodromy, i.e., F_{N}(N^{−kN}z) is a hypergeometric function with only 1’s as lower parameters. Its other solutions around z = 0 can then be obtained by Frobenius’ method, see [29].) The function

q_{N}(z) := z exp(G_{N}(z)/F_{N}(z)) ∈ zℚ[[z]]  (1.1)

is usually called the canonical coordinate, and its compositional inverse z_{N}(q) is the prototype of a mirror map. In this paper, by abuse of terminology, we will also use the term “mirror map” for any canonical coordinate. (^{2})

The case N = 1 is trivial because G_{1}(z) = 0 for any k. When N = 2 and k = 1, we have F_{2}(z) = 1/\sqrt{1 − 4z} and q_{2}(z) = 4z/(1 + \sqrt{1 − 4z})^{2}. For small values of N and k, modular forms rapidly enter the picture. Indeed, for N = 2 and k = 2, the compositional inverse of q_{2}(z) is equal to λ(q)/16 := q ∏_{n=1}^{∞} ((1 + q^{2n})/(1 + q^{2n−1}))^{8}, which is a modular function of the variable τ, with ℑ(τ) > 0, defined by q = exp(2iπτ). Furthermore, when N = 2 and k = 3, the compositional inverse of q_{2}(z) is equal to λ(q)(1 − λ(q))/16. See for example the discussion in [2, pp. 111–113] (and also for the importance of such facts in Diophantine approximation), and see [19, Sec. 3] for a discussion of the modularity of the case N = 4, k = 1. In fact, Doran [9] proved a result that enables one to describe all mirror maps of modular origin.

1Most of the time, the dependence on k will not be explicit in our notation for the sake of better readability. Furthermore, for the results of the present paper, it is sufficient to consider the series as formal power series.

2Canonical coordinates and mirror maps have distinct geometric meanings. However, in the numbertheoretic study undertaken in the present paper, they play strictly the same role, because (z^{−1}q(z))^{1/τ} ∈ 1+zℤ[[z]] for some integer τ implies that (q^{−1}z(q))^{1/τ} ∈ 1+qℤ[[q]], and conversely. (See [18, Introduction].)

The most famous non-modular example of a mirror map is q_{5}(z) (when N = 5, k = 1), which was used in the epoch-making paper by the physicists Candelas et al. [6]. Without going into details (see also [22, 24, 27] and the references therein), let us give a short explanation of the importance of q_{5}(z). Starting from the family M of quintic hypersurfaces in ℙ^{4}(ℂ) defined by ∑_{k=1}^{5} x_{k}^{5} − 5z ∏_{k=1}^{5} x_{k} = 0 (z being a complex parameter), Candelas et al. naturally associate another family W of manifolds (the “mirror of M”) which turn out to be Calabi–Yau. To W, one can naturally associate a vector of periods (depending on z) which are solutions of the same differential equation (namely, the Picard–Fuchs equation of W). This equation is simply the hypergeometric differential equation satisfied by F_{5}(z) and G_{5}(z) + log(z)F_{5}(z) (case N = 5, k = 1 above). Then they observed the non-trivial property that the Taylor coefficients of q_{5}(z) are integers. Furthermore, let us define the Yukawa coupling (^{3})

K(q) := \frac{5}{1 − 5^{5}z_{5}(q)} · \frac{1}{F_{5}(z_{5}(q))^{2}} · (\frac{qz′_{5}(q)}{z_{5}(q)})^{3} ∈ ℚ[[q]],  (1.2)

where z_{5}(q) is the compositional inverse of q_{5}(z), and write it as K(q) = 5 + ∑_{d=1}^{∞} k_{d} \frac{q^{d}}{1−q^{d}} , which is formally possible. Candelas et al. observed that the instanton number n_{d} := k_{d}/d^{3} is an integer for all d ≥ 1, which is already a non-trivial fact, but that furthermore n_{d} seems to be the number of rational curves of degree d lying on the initial quintic M, thereby providing an effective algorithm to compute these numbers. These striking observations generated much interest amongst algebraic geometers, and this culminated in the work of Givental [12] and Lian et al. [20] where it is proved that, if for a given d the curves of degree d are all rigid, then there are n_{d} of them; see also the discussion in [27, p. 49]. In fact, the coincidence was proved to be true for d ≤ 9, and the first difference occurs at d = 10 (see [8]). (However, we do not address questions related to Yukawa couplings in the present paper.)

1.2. Integrality of mirror maps. Deep results can also be proved without any appeal to algebraic geometry. Indeed, using p-adic methods, Lian and Yau were the first to prove that q_{5}(z) has integral Taylor coefficients. In fact, in [19, Sec. 5, Theorem 5.5], they proved that q_{N}(z) ∈ zℤ[[z]] for k = 1 and any N which is a prime number. Zudilin [30, Theorem 3] extended their result to any k ≥ 1 and any N which is a prime power and made the following conjecture, which is also implicit in the cited articles of Lian and Yau, for example at the end of the introduction of [21]. In fact, the conjecture probably belongs to the folklore of mirror symmetry theory, and it seems that it had been left open for N not a prime power.

Conjecture 1. For any integers k ≥ 1 and N ≥ 1, we have q_{N}(z) ∈ zℤ[[z]].

Such integrality questions for mirror maps and Yukawa couplings undoubtedly remain an important question for algebraic geometers, as is witnessed by the very recent preprint [28] (which is an elaborate version of [16]) on this subject. The mirror maps that are considered in that paper comprise ours. When both approaches apply simultaneously, our results in Theorems 1–5 are stronger than Theorem 2 in [28, Sec. 1.3]. Indeed, we prove that certain mirror maps have integral Taylor coefficients, while in [28] the weaker statement is proved that mirror maps have Taylor coefficients in ℤ[1/n], where n is an integer parameter of geometric origin which is at least 2 (by assumption iii) just before Theorem 2 in [28]). On the other hand, the range of applicability of [28, Theorem 2] is much wider than ours. It is interesting to note that our approach as well as the one in [28] are heavily based on p-adic analysis, so that both are clearly close in spirit, although the exact methods that are applied are different.

3The Yukawa coupling is a geometric object whose definition in a specific situation can be found in [4, Definition 4.5.2]. In the present case, it can be computed as in (1.2).

In [21], Lian and Yau strengthened their result from [19] by proving an observation made by physicists:

(z^{−1}q_{N}(z))^{1/N} ∈ ℤ[[z]]  (1.3)

for k = 1 and any prime N.

Our original goal was to settle Conjecture 1 and to prove (1.3) for arbitrary k and N. In the present paper, we shall accomplish much more: we establish refinements of the above integrality assertions which enable us to prove results that are even stronger and more general than Conjecture 1 or (1.3). In fact, our refinements will go into two different, only barely overlapping directions. One direction is inspired by a conjecture in Zudilin’s paper [30], while the other seems to be entirely new. In the remainder of this introductory section, we describe these two directions, and we present our results. Their proofs will then be given in the subsequent sections.

1.3. Refinements of (1.3), part I. We describe the second direction of refinement of (1.3) first. Starting point is the observation that

∑_{j=1}^{N−1} H(j/N, m) − (N − 1)H(1, m) = NH_{Nm} − NH_{m}.

Hence, with

G_{L,N}(z) := ∑_{m=1}^{∞} H_{Lm} \frac{(Nm)!^{k}}{m!^{kN}} z^{m}

and q_{L,N}(z) := exp(G_{L,N}(z)/F_{N}(z)), we have

q_{N}(z) = zq_{N,N}(z)^{kN}q_{1,N}(z)^{−kN}.  (1.4)

We are ready to state our first refinement of Conjecture 1 and (1.3).

Theorem 1. For any integers k, N ≥ 1 and L ∈ {1, 2, …, N}, we have q_{L,N}(z) ∈ ℤ[[z]].

Clearly, due to (1.4), Theorem 1 implies Conjecture 1 and (1.3).

However, much more is true. For any sequence N = (N_{1}, …, N_{k}) of positive integers (the N_{j}’s are not necessarily distinct) and any integer L ≥ 1, let us define the power series

F_{N}(z) := ∑_{m=0}^{∞} ( ∏_{j=1}^{k} \frac{(N_{j}m)!}{m!^{N_{j}}})z^{m},

G_{L,N}(z) := ∑_{m=1}^{∞} H_{Lm}( ∏_{j=1}^{k} \frac{(N_{j}m)!}{m!^{N_{j}}})z^{m},

and the function q_{L,N}(z) := exp (G_{L,N}(z)/F_{N}(z)). The series F_{N}(z) and G_{N}(z) + log(z)F_{N}(z), where G_{N}(z) is a suitable linear combination of the series G_{L,N}(z) (for different L’s), are solutions of a hypergeometric differential equation with maximal unipotent monodromy (of type (1.18) below). That differential equation is the Picard–Fuchs equation of a one parameter family of mirror manifolds V′ of a complete intersection V of k hypersurfaces V_{1}, …, V_{k} of degrees N_{1}, …, N_{k} in ℙ^{d+k}(ℂ): V is a family of Calabi–Yau manifolds if one chooses d equal to ∑_{j=1}^{k} N_{j} − k − 1. The mirrors V′ are explicitly constructed in [4, Sec. 5.2].

Let us define M_{N} = ∏_{i=1}^{k} N_{i}! and, for L ≥ 1, V_{L,N} as the largest integer such that q_{L,N}(z)^{1/V_{L,N}} ∈ ℤ[[z]]. (^{4}) While we are not able to determine V_{L,N} precisely, we shall prove the following result which, as should become clear from the discussion in Section 10, comes relatively close.

Theorem 2. Let Θ_{L} := L!/ gcd(L!, L! H_{L}) be the denominator of H_{L} when written as a reduced fraction. Then, for any integers N_{1}, …, N_{k} ≥ 1 and L ∈ {1, 2, …, max(N_{1}, …, N_{k})},

we have q_{L,N}(z)^{\frac{Θ_{L}}{M_{N}}} ∈ ℤ[[z]].

Remarks 1. (a) For any integer s ≥ 1, we have q_{L,N}(z)^{1/s} = 1 + s^{−1}M_{N}H_{L}z + O(z^{2}), and hence Theorem 2 is optimal when L = 1. This is not necessarily the case for other values of L and, in particular, Theorem 2 can sometimes be improved when L = N_{1} = ⋯ = N_{k} (see Theorem 3 below).

(b) Considering the case k = 1 and L = N in Theorem 2, the first few values of M_{(N)}/Θ_{N} = gcd(N!, N! H_{N}) (for N ≥ 1) are

1, 1, 1, 2, 2, 36, 36, 144, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, … (1.5)

In the On-Line Encyclopedia of Integer Sequences [23], this sequence is entry A056612.

By forming a suitable product of the functions q_{L,N}(z), Theorem 2 implies the integrality of Taylor coefficients of the corresponding mirror maps q_{N}(z) of the mirror pair (V, V′).

4Let q(z) be a given power series in ℤ[[z]], and let V be the largest integer with the property that q(z)^{1/V} ∈ ℤ[[z]]. Then V carries complete information about all integers with that property: namely, the set of integers U with q(z)^{1/U} ∈ ℤ[[z]] consists of all divisors of V. Indeed, it is clear that all divisors of V belong to this set. Moreover, if U_{1} and U_{2} belong to this set, then also lcm(U_{1}, U_{2}) does (cf. [14, Lemma 5] for a simple proof based on Bézout’s lemma).

According to [4, Prop. 5.1.2], the Yukawa coupling in this case is equal to

K(q) := \frac{N_{1}N_{2} ⋯ N_{k}}{1 − λz_{N}(q)} · \frac{1}{F_{N}(z_{N}(q))^{2}} · (\frac{qz′_{N}(q)}{z_{N}(q)})^{d},

where λ = ∏_{j=1}^{k} N_{j}^{N_{j}} and z_{N}(q) is the compositional inverse of q_{N}(z). When d = 3 = ∑_{j=1}^{k} N_{j} − k − 1, the formal expansion K(q) = K(0) + ∑_{m=1}^{∞} k_{m} \frac{q^{m}}{1−q^{m}} enables one to count rational curves on the Calabi–Yau threefold V, at least for small values of the degree of the curves.

An outline of the proof of Theorem 2 is given in Section 3, with details being filled in in later sections. Since we shall refer to it below, we remark that an alternative way to define the integer Θ_{L} is via

Θ_{L} = ∏_{p≤L}p^{− min{0,v_{p}(H_{L})}},  (1.6)

where v_{p}(α) denotes the p-adic valuation of α. Here and in the sequel of the article, the letter p will always represent a prime number.

Due to (1.4), Theorem 2 has the following consequence for the original mirror map q_{N}(z), thus improving significantly upon (1.3).

Corollary 1. For all integers k ≥ 1 and N ≥ 1, we have

(z^{−1}q_{N}(z))^{\frac{Θ_{N}}{N!^{k}kN}} ∈ ℤ[[z]],

where q_{N}(z) is the mirror map in (1.1).

In particular, in the emblematic case of the mirror map q_{5}(z) of the quintic (case N = 5, k = 1), we obtain that (z^{−1}q_{5}(z))^{1/10} ∈ ℤ[[z]], which improves on (1.3) by a factor of 2.

The next theorem presents the improvement of Theorem 2 for the case L = N_{1} = ⋯ = N_{k} which was announced in Remark 1(a) above.

Theorem 3. Let N be a positive integer, N = (N, N, …, N), with k occurrences of N, and let Ξ_{1} = 1, Ξ_{7} = 1/140, and, for N ∉ {1, 7},

Ξ_{N} := ∏_{p≤N}p^{min{2+ξ(p,N),v_{p}(H_{N})}},  (1.7)

where ξ(p, N) = 1 if p is a Wolstenholme prime (i.e., a prime p for which v_{p}(H_{p−1}) ≥ 3

(^{5})) or N is divisible by p, and ξ(p, N) = 0 otherwise. Then q_{N,N}(z)^{\frac{1}{Ξ_{N} N!^{k}}} ∈ ℤ[[z]].

Remarks 2. For better comprehension, we discuss the meaning of the statement of Theorem 3 and its implications; in particular, we address some fine points of the definition of Ξ_{N}.

5Presently, only two such primes are known, namely 16843 and 2124679, and it is unknown whether there are infinitely many Wolstenholme primes or not.

(a) The case of N = 1 is trivial since q_{1,(1,...,1)}(z) = 1/(1 − z). Furthermore, we have

Ξ_{7} = \frac{1}{140} = 2^{v_{2}(H_{7})}5^{v_{5}(H_{7})}7^{v_{7}(H_{7})},

which differs by a factor of 3 from the right-hand side of (1.7) with N = 7 (since v_{3}(H_{7}) = v_{3}(\frac{363}{140}) = 1).

(b) Since q_{N,N}(z) = 1 + H_{N}N!^{k}z + O(z^{2}), it is clear that q_{N,N}(z)^{1/(p^{vp(H_{N})+1}N!^{k})} ∉ ℤ[[z]], so that the exponent of p in the prime factorisation of V_{N,N} can be at most v_{p}(H_{N}N!^{k}). In Theorem 3, this theoretically maximal exponent is further cut down. First of all, the number Ξ_{N} appearing there contains no prime factor p > N. Moreover, for primes p with p ≤ N and v_{p}(H_{N}) ≥ 3, the definition of Ξ_{N} cuts the theoretically maximal exponent v_{p}(H_{N}N!^{k}) of p down to 2 + v_{p}(N!^{k}) respectively 3 + v_{p}(N!^{k}), depending on whether ξ(p, N) = 0 or ξ(p, N) = 1. In items (c)–(e) below, we address the question of how serious this cut is expected to be.

(c) Clearly, the minimum appearing in the exponent of p in the definition (1.7) of Ξ_{N} is v_{p}(H_{N}) as long as v_{p}(H_{N}) ≤ 2. In other words, the exponent of p in the prime factorisation of Ξ_{N} depends largely on the p-adic behaviour of H_{N}. An extensive discussion of this topic, with many interesting results, can be found in [5]. We have as well computed a table of harmonic numbers H_{N} up to N = 1000000. (^{6}) Indeed, the data suggest that pairs (p, N) with p prime, p ≤ N, and v_{p}(H_{N}) ≥ 3 are not very frequent. More precisely, so far only five examples are known with v_{p}(H_{N}) = 3: four for p = 11, with N = 848, 9338, 10583, and 3546471722268916272, and one for p = 83 with

N = 79781079199360090066989143814676572961528399477699516786377994370\
78839681692157676915245857235055200779421409821643691818  (1.8)

(see [5, p. 289]; the value of N in (1.8), not printed in [5], was kindly communicated to us by David Boyd). There is no example known with v_{p}(H_{N}) ≥ 4. It is, in fact, conjectured that no p and N exist with v_{p}(H_{N}) ≥ 4. Some evidence for this conjecture (beyond mere computation) can be found in [5].

(d) Since in all the five examples for which v_{p}(H_{N}) = 3 we neither have p | N (the gigantic number in (1.8) is congruent to 42 mod 83) nor that the prime p is a Wolstenholme prime, the exponent of p in the prime factorisation of Ξ_{N} in these cases is 2 instead of v_{p}(H_{N}) = 3.

(e) On the other hand, should there be a prime p and an integer N with p ≤ N, v_{p}(H_{N}) ≥ 3, p a Wolstenholme prime or p | N, then the exponent of p in the prime factorisation of Ξ_{N} would be 3. However, no such examples are known. We conjecture that there are no such pairs (p, N). If this conjecture should turn out to be true, then, given N ∉ {1, 7}, the definition of Ξ_{N} in (1.7) could be simplified to

Ξ_{N} := ∏_{p≤N}p^{min{2,v_{p}(H_{N})}}.  (1.9)

6The summary of the table is available at http://www.mat.univie.ac.at/~kratt/artikel/H.html.

In view of (1.6) and the fact that M_{N} = N!^{k} for the vector N in Theorem 3, this theorem improves upon Theorem 2 in the case L = N. Namely, Theorem 3 is always at least as strong as Theorem 2, and it is strictly stronger if N ≠ 7 and v_{p}(H_{N}) ≥ 1 for some prime p less than or equal to N. Indeed, the smallest N ≠ 7 with that property is N = 20, in which case v_{5}(H_{20}) = 1.

We remark that strengthenings of Theorem 2 in the spirit of Theorem 3 for more general choices of the parameters can also be obtained by our techniques but are omitted here.

We outline the proof of Theorem 3 in Section 8, with the details being filled in in the subsequent Section 9. As we explain in Section 10, we conjecture that Theorem 3 cannot be improved if k = 1, that is, that for k = 1 the largest integer t_{N} such that q_{N,N}(z)^{1/t_{N}} ∈ ℤ[[z]] is exactly Ξ_{N}N!. Propositions 2 and 3 in Section 10 show that this conjecture would immediately follow if one could prove the conjecture from Remarks 2(c) above that there are no primes p and integers N with v_{p}(H_{N}) ≥ 4.

Even if the series q_{N,N}(z) appears in the identity (1.4), relating our mirror map q_{N}(z) to the series q_{L,N}(z) (with L = 1 and L = N), Theorem 3 does not imply an improvement over Corollary 1, for the coefficient of z in (z^{−1}q_{N}(z))^{Θ_{N} /(pN!^{k}kN)} is equal to \frac{Θ_{N}}{p} (H_{N} − 1), and, thus, it will not be integral for primes p with v_{p}(H_{N}) > 0. Still, there is an improvement of Corollary 1 in the spirit of Theorem 3. It involves primes p with v_{p}(H_{N} − 1) > 0 instead.

Theorem 4. Let N be a positive integer with N ≥ 2, and let

Ω_{N} := ∏_{p≤N}p^{min{2+ω(p,N),v_{p}(H_{N} −1)}},  (1.10)

where ω(p, N) = 1 if p is a Wolstenholme prime or N ≡ ±1 mod p, and ω(p, N) = 0

otherwise. Then (z^{−1}q_{N}(z))^{\frac{1}{Ω_{N} N!^{k}kN}} ∈ ℤ[[z]].

Remarks 3. Also here, some remarks are in order to get a better understanding of the above theorem.

(a) If v_{p}(H_{N}) < 0, then v_{p}(H_{N}) = v_{p}(H_{N} − 1). Hence, differences in the prime factorisations of Ξ_{N} and Ω_{N} can only arise for primes p with v_{p}(H_{N}) ≥ 0.

(b) Since z^{−1}q_{N}(z) = 1 + (H_{N} − 1)N!^{k}kNz + O(z^{2}), it is clear that

(z^{−1}q_{N}(z))^{1/(p^{vp(H_{N} −1)+1}N!^{k}kN)} ∉ ℤ[[z]].

If Ṽ_{N} denotes the largest integer such that (z^{−1}q_{N}(z))^{1/Ṽ_{N} kN} is a series with integer coefficients, the exponent of p in Ṽ_{N} can be at most v_{p}((H_{N} − 1)N!^{k}). In Theorem 4, this theoretically maximal exponent is further cut down. Namely, as is the case for Ξ_{N}, the number Ω_{N} in (1.10) contains no prime factor p > N. Moreover, for primes p with p ≤ N and v_{p}(H_{N} − 1) ≥ 3, the definition of Ω_{N} cuts the theoretically maximal exponent v_{p}((H_{N} − 1)N!^{k}) of p down to 2 + v_{p}(N!^{k}) respectively 3 + v_{p}(N!^{k}), depending on whether ω(p, N) = 0 or ω(p, N) = 1.

(c) Concerning the question whether there are at all primes p and integers N with high values of v_{p}(H_{N} − 1), we are not aware of any corresponding literature. Our table of harmonic numbers H_{N} mentioned in Remarks 2(c) does not contain any pair (p, N) with v_{p}(H_{N} − 1) ≥ 3. (^{7}) In “analogy” to the conjecture mentioned in Remarks 2(c), we conjecture that no p and N exist with v_{p}(H_{N} − 1) ≥ 4. It may even be true that there are no p and N with v_{p}(H_{N} − 1) ≥ 3, in which case the definition of Ω_{N} in (1.10) could be simplified to

Ω_{N} := ∏_{p≤N}p^{min{2,v_{p}(H_{N} −1)}}.  (1.11)

In view of (1.6), Theorem 4 improves upon Corollary 1. Namely, Theorem 4 is always at least as strong as Corollary 1, and it is strictly stronger if v_{p}(H_{N} − 1) ≥ 1 for some prime p less than or equal to N. The smallest N with that property is N = 21, in which case v_{5}(H_{21} − 1) = 1.

We sketch the proof of Theorem 4 in Section 11. We also explain in that section that we conjecture that Theorem 4 with k = 1 is optimal, that is, that for k = 1 the largest integer u_{N} such that (z^{−1}q_{N}(z))^{\frac{1}{Nu_{N}}} ∈ ℤ[[z]] is exactly Ω_{N}N!. Propositions 4 and 5 in Section 11 show that this conjecture would immediately follow if one could prove the conjecture from Remarks 3(c) above that there are no primes p and integers N with v_{p}(H_{N} − 1) ≥ 4. As a matter of fact, the sequence (u_{2N})_{N≥1} appears also in the On-Line Encyclopedia of Integer Sequences [23], as sequence A007757, contributed by R. E. Borcherds under the denomination “Dwork–Kontsevich sequence” around 1995, without any reference or explicit formula for it, however. (^{8})

1.4. Refinements of (1.3), part II. We now move on to describe the other direction of refinement of Conjecture 1 and (1.3), inspired by Zudilin’s paper [30]. For any given integer N ≥ 1, let r_{1}, r_{2}, …, r_{d} denote the integers in {1, 2, …, N} which are coprime to N. It is well-known that d = φ(N), Euler’s totient function, which is given by φ(N) = N_{∏p|N}(1 − \frac{1}{p}). Set C_{N} := N^{φ(N)}_{∏p|N}p^{φ(N)/(p−1)}, which is an integer because p − 1 divides φ(N) for any prime p dividing N. Let us also define the Pochhammer symbol (α)_{m} for complex numbers α and non-negative integers m by (α)_{m} := α(α + 1) ⋯ (α + m − 1) if m ≥ 1 and (α)_{0} := 1. It can be proved (see [30, Lemma 1], or (1.15) together with Lemma 28(iii)) that, for any integer m ≥ 0,

B_{N}(m) := C_{N}^{m} ∏_{j=1}^{φ(N)} \frac{(r_{j}/N)_{m}}{m!}  (1.12)

is an integer. We will as well use B_{N}(m) := ∏_{j=1}^{k} B_{N_{j}}(m) for vectors N = (N_{1}, …, N_{k}) of positive integers. Zudilin also established another representation for B_{N}(m) (see [30, Lemma 4]). Namely, let p_{1}, p_{2}, …, p_{ℓ} denote the distinct prime factors of N, and let us define the vectors of integers

7The summary of the corresponding table, containing pairs (p, N) with p ≤ N and v_{p}(H_{N} − 1) > 0, is available at http://www.mat.univie.ac.at/~kratt/artikel/H1.html.

8In private communication, both, Borcherds and Kontsevich could not remember where exactly this sequence and its denomination came from.

(α_{j})_{j=1,...,μ} := (N, \frac{N}{p_{j_{1}}p_{j_{2}}} , \frac{N}{p_{j_{1}}p_{j_{2}}p_{j_{3}}p_{j_{4}}} , …)_{1≤j_{1}<j_{2}<···≤ℓ}  (1.13)

and

(β_{j})_{j=1,...,η} := (\frac{N}{p_{j_{1}}} , \frac{N}{p_{j_{1}}p_{j_{2}}p_{j_{3}}} , …, 1, 1, …, 1)_{1≤j_{1}<j_{2}<···≤ℓ} ,  (1.14)

where α_{1} + α_{2} + ⋯ + α_{μ} = β_{1} + β_{2} + ⋯ + β_{η}. Then we have

B_{N}(m) = \frac{∏_{j=1}^{μ}(α_{j}m)!}{∏_{j=1}^{η}(β_{j}m)!}.  (1.15)

For example, we have

B_{4}(m) = \frac{(4m)!}{(2m)! m!^{2}} , B_{6}(m) = \frac{(6m)!}{(3m)! (2m)! m!}, B_{30}(m) = \frac{(30m)! (5m)! (3m)! (2m)!}{(15m)! (10m)! (6m)! m!^{9}}.

We now define the functions

H_{N}(m) := ∑_{j=1}^{φ(N)} H(r_{j}/N, m) − φ(N)H(1, m),

F_{N}(z) := ∑_{m=0}^{∞} ( ∏_{j=1}^{k} B_{N_{j}}(m))z^{m},  (1.16)

and

G_{N}(z) := ∑_{m=1}^{∞} ( ∑_{j=1}^{k} H_{N_{j}}(m))( ∏_{j=1}^{k} B_{N_{j}}(m))z^{m}.

We can now state Zudilin’s conjecture from [30, p. 605].

Conjecture 2 (Zudilin). For any positive integers N_{1}, N_{2}, …, N_{k}, we have q_{N}(z) := z exp(G_{N}(z)/F_{N}(z)) ∈ zℤ[[z]].

As explained by Zudilin, this conjecture implies Conjecture 1 because any function q_{N}(z) is equal to q_{N}(z) for a suitable N. More precisely, for a given N ≥ 1, the vector N to choose consists of all the positive divisors of N but 1. For example, we have q_{9}(z) = q_{(9,3)}(z), q_{12}(z) = q_{(12,6,4,3,2)}(z) and q_{35}(z) = q_{(7,5)}(z). However, Conjecture 2 does not imply any of the previous theorems. Zudilin proved that his conjecture holds under the condition that if a prime number divides N_{1}N_{2} ⋯ N_{k} then it also divides each N_{j}. This applies to the function q_{(9,3)}(z), but neither to q_{(12,6,4,3,2)}(z) nor to q_{(7,5)}(z), for example.

We claim that Conjecture 2 follows from the theorem below. For the statement of the theorem, for an integer L ≥ 1, we need to define

G_{L,N}(z) := ∑_{m=1}^{∞} H_{Lm}( ∏_{j=1}^{k} B_{N_{j}}(m))z^{m}.

Theorem 5. For any integers N_{1}, N_{2}, …, N_{k} ≥ 1 and L ∈ {1, 2, …, max(N_{1}, …, N_{k})}, we have q_{L,N}(z) := exp(G_{L,N}(z)/F_{N}(z)) ∈ ℤ[[z]].

An outline of the proof of this theorem is given in Section 13, with details being filled in in subsequent sections.

To see that Theorem 5 implies Conjecture 2, we prove in Lemma 27 in Section 14 that, for a given N, we have

H_{N}(m) = ∑_{j=1}^{μ} α_{j}H_{α_{j}m} − ∑_{j=1}^{η} β_{j}H_{β_{j}m}.  (1.17)

Clearly, the α_{j}’s and β_{j}’s (defined in (1.13) and (1.14)) are less than N. Therefore ∑_{j=1}^{k} H_{N_{j}}(m) is a finite sum of terms of the form λH_{Lm}, where λ and L are integers with L ∈ {1, 2, …, max(N_{1}, …, N_{k})}, and the claimed implication follows.

It is proved in [26] that there are exactly fourteen mirror maps q_{N}(z) associated to complete intersections Calabi–Yau threefolds in weighted projective spaces: q_{(12)}(z), q_{(5)}(z), q_{(8)}(z), q_{(10)}(z), q_{(3,3)}(z), q_{(4,2,2)}(z), q_{(2,2,2,2)}(z), q_{(4,4)}(z), q_{(6,6)}(z), q_{(4,3)}(z), q_{(6,2,2)}(z), q_{(3,2,2)}(z), q_{(3,6)}(z), q_{(6,4)}(z). All but the first one were already described in [4, Sec. 6.4]. Theorem 5 implies that they all have integral Taylor coefficients, which is a new result for the last five of them.

More generally, for any N, the functions F_{N}(z) and G_{N}(z) + log(z)F_{N}(z) satisfy a hypergeometric differential equation Ly = 0 with maximal unipotent monodromy at the origin, where the differential operator L is defined by

L := (z \frac{d}{dz} )^{φ(N_{1})+···+φ(N_{k})} − zC_{N} ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} (z \frac{d}{dz} + \frac{r_{i,j}}{N_{j}} ) .  (1.18)

Here, C_{N} = C_{N_{1}}C_{N_{2}} ⋯ C_{N_{k}} and the r_{i,j} ∈ {1, 2, …, N_{j}} form the residue classes mod N_{j} which are coprime to N_{j}. This equation is the Picard–Fuchs equation of the mirror Calabi–Yau family of a one parameter family of Calabi–Yau complete intersections in a weighted projective space (see [7] and [15, Sec. 3]).

It is natural to expect refinements of Theorem 5 in the spirit of Theorem 2 and Theorem 4. We did not make a systematic research in this direction, but it could be interesting to do so. For example, in the case k = 1 and N = (6), it seems that the following relations are best possible: q_{1,(6)}(z)^{1/60}, q_{2,(6)}(z)^{1/6}, q_{3,(6)}(z)^{1/2}, q_{4,(6)}(z), q_{5,(6)}(z) and q_{6,(6)}(z) are in ℤ[[z]].

As a first step towards such refinements, we prove in Lemma 30 in Section 14 that B_{N}(1) always divides B_{N}(m) for any m ≥ 1 and any N. Our techniques enable us to deduce that q_{1,N}(z)^{1/B_{N}(1)} ∈ ℤ[[z]], which proves the above assertion that q_{1,(6)}(z)^{1/60} ∈ ℤ[[z]]. In fact, this is optimal because q_{1,N}(z) = 1 + B_{N}(1)z + O(z^{2}). It turns out that B_{N}(1) is a natural generalisation of the quantity M_{N} which appears in Theorem 2. However, for larger values of the parameter L, we do not know what the analogue of the quantity M_{N}/Θ_{L} appearing in Theorem 2 would be.

Let us consider the hypergeometric functions F with rational parameters for which there exists a constant C = C(F) > 0 such that the Taylor coefficients of F (Cz) are of the form

\frac{∏_{j=1}^{r}(a_{j}m)!}{∏_{j=1}^{s}(b_{j}m)!},  (1.19)

where m, the a_{j}’s and b_{j}’s are non-negative integers, and ∑_{j=1}^{r} a_{j} = ∑_{j=1}^{s} b_{j}. A well-known theorem of Landau (cf. [26, Proposition 1.1]) determines the quotients of the form (1.19) which are integer-valued for all integers m ≥ 0. The quantities B_{N}(m) are such examples but do not describe all these quotients, as demonstrated by the example (3m)!/(m! (2m)!).

In our context, it is therefore natural to ask the following question: what are the hypergeometric functions F (with z changed to Cz for a suitable C) such that

(i) their Taylor coefficients are integers and of the form (1.19), (ii) the associated “mirror map” type functions (that is, functions defined in a manner

analogous to q_{N}(z)) have integral Taylor coefficients? Of course, Condition (ii) has a sense only if the underlying differential equation has two

solutions F (z) and G(z) + log(z)F (z) with F (z) and G(z) holomorphic at 0, which in turn implies that at least one of the lower parameters of F is equal to 1. Rodriguez–Villegas proved in [26] a result which, once translated in our setting, says that the functions F with maximal unipotent monodromy (i.e., with only 1’s as lower parameters) and satisfying (i) are exactly the functions F_{N} defined by (1.16). Theorem 5 proves that such functions also satisfy (ii). Numerical experiments seem to indicate that there are many examples of such functions F which are not of the form (1.16).

1.5. Structure of the paper. We now briefly review the organisation of the rest of the paper. Following the steps of previous authors, our approach for proving Theorems 1–5 uses p-adic analysis. In particular, we make essential use of Dwork’s theory of formal congruences, which we recall in Section 2. Since the details of our proofs are involved, we provide brief outlines of the proofs of Theorems 2–5 (with Theorem 2 implying Theorem 1) in separate sections. Namely, Section 3 provides an outline of the proof of Theorem 2, while Section 13 provides an outline of the proof of Theorem 5. In both cases, the proof is reduced to a certain number of lemmas. The lemmas which are necessary for the proof of Theorem 2 are established in Sections 4–7, while those necessary for the proof of Theorem 5 are established in Sections 14–16. Finally, Section 8 contains an outline of the proof of Theorem 3, and Section 11 contains a sketch of the proof of Theorem 4. Both are largely based on arguments already used in the proof of Theorem 2. The outline of the proof of Theorem 3 requires again several auxiliary lemmas, whose proofs are postponed to a separate section, Section 9. The subsequent section, Section 10, reports on the evidence to believe (or not to believe) that the value t_{N} (defined in the next-to-last paragraph before Theorem 4) is given by t_{N} = Ξ_{N}N!, N = 1, 2, …, while Section 12 addresses the question of whether the Dwork–Kontsevich sequence (u_{N})_{N≥1} (defined in the last paragraph in Section 1.3) is (or is not) given by u_{N} = Ω_{N}N!, N = 1, 2, . . ..

We draw the reader’s attention to the fact that, while the general line of our approach follows that of previous authors (particularly [21]), there does arise a crucial difference (other than just technical complications): the reduction and rearrangement of the sums C(a + Kp) in Sections 3 and 8, respectively C(a + Kp) in Section 13, via the congruence (3.3) requires a new reduction step, namely Lemma 5, congruence (8.5), and Lemma 24, respectively. In fact, the proofs of these two lemmas and the congruence form the most difficult parts of our paper. (In previous work, the use of (3.3) sufficed because the corresponding authors restricted themselves to N being prime or a prime power.)

We expect that (variations of) the detailed techniques that we present here can also serve to prove integrality results for other types of mirror maps, and in particular some of Observations 1–7 in [1, Sec. 4]. We also hope that our methods will turn out to be useful in the context of the very general multivariable mirror maps coming from the Gelfand– Kapranov–Zelevinsky hypergeometric series: see [4, Sec. 7.1], [15] and [25, Sec. 8] for numerous examples related to Calabi–Yau manifolds which are complete intersections in products of weighted projective spaces.

2. Dwork’s theory of formal congruences

Since the Taylor coefficients of F_{N}(z) and G_{L,N}(z) (respectively F_{N}(z) and G_{L,N}(z)) are explicit, we could also obtain an explicit formula for the coefficients of q_{L,N}(z) (respectively q_{L,N}(z)). Unfortunately, such a formula does not seem to be very useful to prove the desired integrality properties. Instead, we will follow the authors of the articles cited in the Introduction, who all relied on Dwork’s theory.

First, consider a formal power series S(z) ∈ ℚ[[z]] and suppose that we want to prove that S(z) ∈ ℤ[[z]].

Lemma 1. Let S(z) be a power series in ℚ[[z]]. If S(z) ∈ ℤ_{p}[[z]] for any prime number p, then S(z) ∈ ℤ[[z]].

This is a consequence of the fact that, given x ∈ ℚ, we have x ∈ ℤ if and only if x ∈ ℤ_{p} for all prime numbers p. Hence we can work in ℚ_{p} for any fixed prime p.

Lemma 2 (“Dwork’s Lemma”). Let S(z) ∈ 1 + zℚ_{p}[[z]]. Then, we have S(z) ∈ 1 + zℤ_{p}[[z]] if and only if

\frac{S(z^{p})}{S(z)^{p}} ∈ 1 + pzℤ_{p}[[z]].

Proof. The proof is neither difficult nor long and can for example be found in the book of Lang [17, Ch. 14, p. 76]. Lang attributes this lemma to Dieudonné and Dwork. □

We now suppose that S(z) = exp(T (z)/τ) for some T (z) ∈ zℚ[[z]] and some integer τ ≥ 1. Dwork’s Lemma implies the following result: τ being any fixed positive integer, we have exp (T (z)/τ) ∈ 1 + zℤ_{p}[[z]] if and only if T (z^{p}) − pT (z) ∈ pτzℤ_{p}[[z]]. (See [19, Corollary 6.7] for a proof.) Since we will be interested in the case when T (z) = g(z)/f(z) with f(z) ∈ 1 + zℤ[[z]] and g(z) ∈ zℚ[[z]], we state this result as follows.

Lemma 3. Given two formal power series f(z) ∈ 1 + zℤ[[z]] and g(z) ∈ zℚ[[z]] and an integer τ ≥ 1, we have exp (g(z)/(τf(z))) ∈ 1 + zℤ_{p}[[z]] if and only if

f(z)g(z^{p}) − p f(z^{p})g(z) ∈ pτzℤ_{p}[[z]].  (2.1)

Because of the special form of the functions which will play the role of f(z) and g(z), we will be able to deduce (2.1) from the following crucial result, also due to Dwork (see [11, Theorem 1.1]). We state it in slightly bigger generality than necessary (i.e., our functions will be independent of r).

Proposition 1 (“Dwork’s Formal Congruences Theorem”). For all integers r ≥ 0, let A_{r} : ℤ_{≥0} → ℚ_{p}^{×}, g_{r} : ℤ_{≥0} → ℤ_{p} \ {0} be mappings such that

(i) |A_{r}(0)|_{p} = 1;

(ii) A_{r}(m) ∈ g_{r}(m)ℤ_{p};

(iii) for all integers u, v, n, r, s ≥ 0 such that 0 ≤ u < p^{s} and 0 ≤ v < p, we have

\frac{A_{r}(v + up + np^{s+1})}{A_{r}(v + up)} − \frac{A_{r+1}(u + np^{s})}{A_{r+1}(u)} ∈ p^{s+1} \frac{g_{r+s+1}(n)}{g_{r}(v + up)} ℤ_{p}.  (2.2)

Furthermore, let F (z) = ∑_{m=0}^{∞} A_{0}(m)z^{m}, G(z) = ∑_{m=0}^{∞} A_{1}(m)z^{m}, and

F_{m,s}(z) = ∑_{j=mp^{s}}^{(m+1)p^{s}−1} A_{0}(j)z^{j}, G_{m,s}(z) = ∑_{j=mp^{s}}^{(m+1)p^{s}−1} A_{1}(j)z^{j}.

Then, for any integers m, s ≥ 0, we have

G(z^{p})F_{m,s+1}(z) − F (z)G_{m,s}(z^{p}) ∈ p^{s+1}g_{s}(m)ℤ_{p}[[z]],  (2.3)

or, equivalently,

∑_{j=mp^{s}}^{(m+1)p^{s}−1} (A_{0}(a + jp)A_{1}(K − j) − A_{1}(j)A_{0}(a + (K − j)p)) ∈ p^{s+1}g_{s}(m)ℤ_{p}  (2.4)

for all a and K with 0 ≤ a < p and K ≥ 0.

Remarks 4. (a) Dwork proved his theorem with A_{r} : ℤ_{≥0} → ℂ_{p}^{×}, g_{r} : ℤ_{≥0} → O_{p} \ {0} and A_{r}(m) ∈ g_{r}(m)O_{p} (where O_{p} is the ring of integers in ℂ_{p}). He obtained a result similar to (2.3) and (2.4), with O_{p} instead of ℤ_{p}. In our more restrictive setting, (2.3) and (2.4) hold because (p^{s+1}g_{s}(m)O_{p}) ∩ ℚ_{p} = p^{s+1}g_{s}(m)ℤ_{p}.

(b) For any integers a and K with 0 ≤ a < p and K ≥ 0, the sum

∑_{j=mp^{s}}^{(m+1)p^{s}−1} (A_{0}(a + jp)A_{1}(K − j) − A_{1}(j)A_{0}(a + (K − j)p))  (2.5)

is exactly the (a + pK)-th Taylor coefficient of G(z^{p})F_{m,s+1}(z) − F (z)G_{m,s}(z^{p}), which explains the equivalence between the formal congruence (2.3) and the congruence (2.4). Note that in (2.5) the value of A_{0} and A_{1} at negative integers must be taken as 0.

(c) Most authors chose g_{s}(m) = 1 or a constant in m and s. We will use instead g_{s}(m) = A_{s}(m): this choice has already been made by Dwork in [10, Sec. 2, p. 37].

(d) Dwork also applied his methods to the problems considered in the present paper. Indeed, he proved a result, namely [11, p. 311, Theorem 4.1], which implies that for any prime p that does not divide N_{1}N_{2} ⋯ N_{k}, the mirror maps q_{N}(z) have Taylor coefficients in ℤ_{p} (see [30, Proposition 2] for details). This fact was used by the authors of [18, 19, 30] who focussed essentially on the remaining case when p divides N_{1}N_{2} ⋯ N_{k}. Our approach is different, for we make no distinction of this kind between prime numbers.

3. Outline of the proof of Theorem 2

In this section, we provide a brief outline of the proof of Theorem 2, reducing it to Lemmas 5–7. These lemmas are subsequently proved in Sections 5–7, with two auxiliary lemmas being the subject of the subsequent section.

By Dwork’s Lemma (or rather its consequence given in Lemma 3), we want to prove that

F_{N}(z)G_{L,N}(z^{p}) − pF_{N}(z^{p})G_{L,N}(z) ∈ p\frac{M_{N}}{Θ_{L}} zℤ_{p}[[z]].

We follow the presentation in [21] and we let 0 ≤ a < p and K ≥ 0.

The (a + Kp)-th Taylor coefficient of F_{N}(z)G_{L,N}(z^{p}) − pF_{N}(z^{p})G_{L,N}(z) is

C(a + Kp) := ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{L(K−j)} − pH_{La+Ljp}),  (3.1)

where B_{N}(m) = ∏_{j=1}^{k} B_{N_{j}}(m) with B_{N}(m) := \frac{(Nm)!}{m!^{N}} (not to be confused with B_{N}(m) and B_{N}(m)). In view of Lemma 3, proving Theorem 2 is equivalent to proving that

C(a + Kp) ∈ p\frac{M_{N}}{Θ_{L}} ℤ_{p}  (3.2)

for all primes p and non-negative integers a and K with 0 ≤ a < p.

The following simple lemma will be frequently used in the sequel.

Lemma 4. For all integers m ≥ 1 and N ≥ 1, we have

B_{N}(m) ∈ N! ℤ.

Proof. Set U_{m}(N) = \frac{(Nm)!}{m!^{N} N!}. For any m, N ≥ 1, we have the trivial relation

U_{m}(N + 1) = \binom{Nm + m − 1}{m − 1}U_{m}(N).

Therefore, since U_{m}(1) = 1, the result follows by induction on N. □

We deduce in particular that B_{N}(m) ∈ M_{N}ℤ for any m ≥ 1.

Since

H_{J} = ∑_{j=1}^{⌊J/p⌋} \frac{1}{pj} +_{∑_{j=1}^{J} ; p∤j} \frac{1}{j},

we have

pH_{J} ≡ H_{⌊J/p⌋} mod pℤ_{p}.  (3.3)

Applying this to J = La + Ljp, we get

pH_{La+Ljp} ≡ H_{⌊La/p⌋+Lj} mod pℤ_{p}.

By Lemma 4, we infer

C(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{L(K−j)} − H_{⌊La/p⌋+Lj}) mod pM_{N}ℤ_{p}. (3.4)

Indeed, if K ≥ 1 or a ≥ 1, this is because a + jp and K − j cannot be simultaneously zero and therefore at least one of B_{N}(a + jp) or B_{N}(K − j) is divisible by M_{N} by Lemma 4. In the remaining case K = a = j = 0, we note that the difference of harmonic numbers in (3.4) is equal to 0, and therefore the congruence (3.4) holds trivially because C(0) = 0.

We now want to transform the sum on the right-hand side of (3.4) to a more manageable expression. In particular, we want to get rid of the floor function ⌊La/p⌋. In order to achieve this, we will prove the following lemma in Section 5.

Lemma 5. For any prime p, non-negative integers a and j with 0 ≤ a < p, positive integers N_{1}, N_{2}, …, N_{k}, and L ∈ {1, 2, …, max(N_{1}, …, N_{k})}, we have

B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) ∈ p\frac{M_{N}}{Θ_{L}} ℤ_{p}.  (3.5)

It follows from Eq. (3.4) and Lemma 5 that

C(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{L(K−j)} − H_{Lj}) mod p\frac{M_{N}}{Θ_{L}} ℤ_{p} ,

which can be rewritten as

C(a+Kp) ≡ − ∑_{j=0}^{K} H_{Lj}(B_{N}(a+jp)B_{N}(K −j)−B_{N}(j)B_{N}(a+(K −j)p)) mod p\frac{M_{N}}{Θ_{L}} ℤ_{p} .

(3.6)

We now use a combinatorial lemma due to Dwork (see [11, Lemma 4.2]) which provides an alternative way to write the sum on the right-hand side of (3.6): namely, we have

∑_{j=0}^{K} H_{Lj}(B_{N}(a + jp)B_{N}(K − j) − B_{N}(j)B_{N}(a + (K − j)p)) = ∑_{s=0}^{r} ∑_{m=0}^{p^{r+1−s}−1} Y_{m,s}, (3.7)

where r is such that K < p^{r}, and

Y_{m,s} := (H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}})S(a, K, s, p, m),

the expression S(a, K, s, p, m) being defined by

S(a, K, s, p, m) := ∑_{j=mp^{s}}^{(m+1)p^{s}−1} (B_{N}(a + jp)B_{N}(K − j) − B_{N}(j)B_{N}(a + (K − j)p)).

In this expression for S(a, K, s, p, m), it is assumed that B_{N}(n) = 0 for negative integers n.

It would suffice to prove that

Y_{m,s} ∈ p\frac{M_{N}}{Θ_{L}} ℤ_{p}  (3.8)

because (3.6) and (3.7) would then imply that C(a + Kp) ∈ p\frac{M_{N}}{Θ_{L}} ℤ_{p}, as desired.

We will prove (3.8) in the following manner. The expression for S(a, K, s, p, m) is of the form considered in Proposition 1. The proposition will enable us to prove the following fact in Section 6.

Lemma 6. For all primes p and non-negative integers a, m, s, K with 0 ≤ a < p, we have

S(a, K, s, p, m) ∈ p^{s+1}B_{N}(m)ℤ_{p}.  (3.9)

Furthermore, in Section 7 we shall prove the following lemma.

Lemma 7. For all primes p, non-negative integers m, positive integers N_{1}, N_{2}, …, N_{k}, and L ∈ {1, 2, …, max(N_{1}, …, N_{k})}, we have

B_{N}(m)(H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}}) ∈ \frac{M_{N}}{p^{s}Θ_{L}} ℤ_{p} .  (3.10)

It is clear that (3.9) and (3.10) imply (3.8). This completes the outline of the proof of Theorem 2.

4. Two auxiliary lemmas

The proof of Lemma 6 in Section 6 requires two further auxiliary results, given in Lemmas 8 and 9 below, the proofs of which form the contents of this intermediary section. In the proof of the first lemma, and also in later proofs, we shall frequently make use of Legendre’s formula for the p-adic valuation of n!, where n is a non-negative integer, which we recall here for convenience:

v_{p}(n!) = ∑_{k=1}^{∞} ⌊\frac{n}{p^{k}} ⌋ .  (4.1)

Lemma 8. For all integers N ≥ 1, n ≥ 0, s ≥ 0, 0 ≤ u < p^{s}, p prime, we have

\frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ B_{N}(n) ℤ_{p},

where B_{N}(m) is defined after (3.1).

Proof. We may rewrite the quotient in the assertion of the lemma as

\frac{(Nu + Nnp^{s})!}{(u + np^{s})!^{N}} · \frac{u!^{N}}{(Nu)!} = \frac{(Nu + Nnp^{s})!}{(Nu)! (Nnp^{s})!} · \frac{(Nnp^{s})!}{(np^{s})!^{N}} · \frac{(np^{s})!^{N}u!^{N}}{(u + np^{s})!^{N}}
= \binom{Nu + Nnp^{s}}{Nu} · \frac{(Nnp^{s})!}{(np^{s})!^{N}} · \binom{u + np^{s}}{u}^{−N}.  (4.2)

The binomial coefficient \binom{Nu+Nnp^{s}}{Nu} is an integer. For the multinomial coefficient we observe that

v_{p}((Nnp^{s})!/(np^{s})!^{N}) = ∑_{k=1}^{∞} (⌊\frac{Nnp^{s}}{p^{k}} ⌋ − N ⌊\frac{np^{s}}{p^{k}} ⌋)

= ∑_{k=1}^{∞} (⌊\frac{Nn}{p^{k}}⌋ − N ⌊\frac{n}{p^{k}} ⌋)

= v_{p}((Nn)!/n!^{N}),

which means that

\frac{(Nnp^{s})!}{(np^{s})!^{N}} ∈ \frac{(Nn)!}{n!^{N}} ℤ_{p}.

Hence, to conclude the proof of the lemma, it suffices to show that

v_{p}(\binom{u + np^{s}}{u}) = 0.  (4.3)

In order to do so, we start with the formula

v_{p}(\binom{u + np^{s}}{u}) = ∑_{k=1}^{∞} ( ⌊\frac{u + np^{s}}{p^{k}} ⌋ − ⌊\frac{u}{p^{k}} ⌋ − ⌊\frac{np^{s}}{p^{k}} ⌋ ).

We distinguish between two cases.

(a) k ≤ s. In this case, (np^{s})/p^{k} is an integer and therefore

⌊\frac{u + np^{s}}{p^{k}} ⌋ − ⌊\frac{u}{p^{k}} ⌋ − ⌊\frac{np^{s}}{p^{k}} ⌋ = ⌊\frac{u}{p^{k}} ⌋ + \frac{np^{s}}{p^{k}} − ⌊\frac{u}{p^{k}} ⌋ − \frac{np^{s}}{p^{k}} = 0.

(b) k > s. In this case, we have ⌊u/p^{k}⌋ = 0 because 0 ≤ u < p^{s} < p^{k}.

Hence,

v_{p}(\binom{u + np^{s}}{u}) = ∑_{k=s+1}^{∞} ( ⌊\frac{u + np^{s}}{p^{k}} ⌋ − ⌊\frac{n}{p^{k−s}} ⌋ ).

Since 0 ≤ u/p^{k} < 1, we have

⌊\frac{u + np^{s}}{p^{k}} ⌋ − ⌊\frac{n}{p^{k−s}} ⌋ ∈ \{0, 1\},

so that it suffices to show that the value 1 cannot be attained. Arguing by contradiction, let us assume that, for some n, u, p, s, we have

⌊\frac{u + np^{s}}{p^{k}} ⌋ − ⌊\frac{n}{p^{k−s}} ⌋ = 1.

Writing, for simplicity, X = ⌊\frac{u + np^{s}}{p^{k}} ⌋, our assumption implies that

\frac{n}{p^{k−s}} < X ≤ \frac{u + np^{s}}{p^{k}} .  (4.4)

X being an integer, we thus have

\frac{n}{p^{k−s}} + \frac{1}{p^{k−s}} ≤ X.  (4.5)

Combining the right inequality of (4.4) with (4.5), we deduce that

\frac{n}{p^{k−s}} + \frac{1}{p^{k−s}} ≤ \frac{u}{p^{k}} + \frac{np^{s}}{p^{k}} ,

which means that u ≥ p^{s}. This contradicts the hypothesis that u < p^{s} and finishes the proof. □

During the proof of Lemma 6 in Section 6, we will also use certain properties of the p-adic gamma function Γ_{p}. This function is defined on integers n ≥ 1 by

Γ_{p}(n) = (−1)^{n} ∏_{(k,p)=1^{k=1}}^{n−1} k.

We will not consider its extension to ℚ_{p}. In the following lemma, we collect the results on Γ_{p} that we shall need later on.

Lemma 9. (i) For all integers n ≥ 1, we have

\frac{(np)!}{n!} = (−1)^{np+1}p^{n}Γ_{p}(1 + np).

(ii) For all integers k ≥ 1, n ≥ 1, s ≥ 0, we have

Γ_{p}(k + np^{s}) ≡ Γ_{p}(k) mod p^{s}.

Proof. See [30, Lemma 7] for (i) and [17, p. 71, Lemma 1.1] for (ii). □

5. Proof of Lemma 5

The assertion is trivially true if ⌊La/p⌋ = 0, that is, if 0 ≤ a < p/L. We may hence assume that p/L ≤ a < p from now on. A further assumption upon which we agree without loss of generality for the rest of the proof is that N_{k} = max(N_{1}, …, N_{k}).

5.1. First part: a weak version of Lemma 5. In a first step, we prove that

B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) ∈ pℤ_{p} .  (5.1)

(The reader should note the absence of the term M_{N}/Θ_{L} in comparison with (3.5).)

For the proof of (5.1), we note that the p-adic valuation of B_{N}(a + pj) is equal to

v_{p}(B_{N}(a + pj)) = ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} (⌊\frac{N_{i}(a + pj)}{p^{ℓ}}⌋ − N_{i} ⌊\frac{a + pj}{p^{ℓ}} ⌋) .

Obviously, all the summands in this sum are non-negative, whence, in particular,

v_{p}(B_{N}(a + pj)) ≥ ∑_{ℓ=1}^{∞} (⌊\frac{N_{k}(a + pj)}{p^{ℓ}}⌋ − N_{k} ⌊\frac{a + pj}{p^{ℓ}} ⌋) .  (5.2)

On the other hand, by definition of the harmonic numbers, we have

H_{Lj+⌊La/p⌋} − H_{Lj} = \frac{1}{Lj + 1} + \frac{1}{Lj + 2} + ⋯ + \frac{1}{Lj + ⌊La/p⌋}.

It therefore suffices to show that

v_{p}(B_{N}(a + pj)) ≥ 1 + max_{1≤ε≤⌊La/p⌋} v_{p}(Lj + ε).  (5.3)

The lower bound on the right-hand side of (5.2) can, in fact, be simplified since 0 ≤ a < p; namely, we have

⌊\frac{a + pj}{p^{ℓ}} ⌋ = ⌊\frac{j}{p^{ℓ−1}} ⌋ .  (5.4)

For a given integer ε with 1 ≤ ε ≤ ⌊La/p⌋, let Lj + ε = p^{d}β, where d = v_{p}(Lj + ε). If we use this notation in (5.2), together with (5.4), we obtain

v_{p}(B_{N}(a + pj)) ≥ ∑_{ℓ=1}^{∞} (⌊\frac{N_{k}a}{p^{ℓ}} − \frac{N_{k}ε}{Lp^{ℓ−1}} + \frac{N_{k}β}{L} p^{d+1−ℓ}⌋ − N_{k} ⌊−\frac{ε}{Lp^{ℓ−1}} + \frac{β}{L}p^{d+1−ℓ}⌋) .

(5.5)

Since ε ≤ ⌊La/p⌋, we have \frac{N_{k}a}{p^{ℓ}} − \frac{N_{k}ε}{Lp^{ℓ−1}} ≥ 0, whence

⌊\frac{N_{k}a}{p^{ℓ}} − \frac{N_{k}ε}{Lp^{ℓ−1}} + \frac{N_{k}β}{L} p^{d+1−ℓ}⌋ ≥ ⌊\frac{N_{k}β}{L} p^{d+1−ℓ}⌋ .  (5.6)

Clearly, we also have

⌊−\frac{ε}{Lp^{ℓ−1}} + \frac{β}{L}p^{d+1−ℓ}⌋ ≤ ⌊\frac{β}{L}p^{d+1−ℓ}⌋ .  (5.7)

If we use (5.6) and (5.7) in (5.5), then we obtain

v_{p}(B_{N}(a + pj)) ≥ ∑_{ℓ=1}^{∞} (⌊\frac{N_{k}β}{L} p^{d+1−ℓ}⌋ − N_{k} ⌊\frac{β}{L}p^{d+1−ℓ}⌋) .  (5.8)

Now we claim that βp^{d+1−ℓ}/L cannot be an integer. Indeed, if it were, then Lγp^{ℓ−1} = βp^{d} = Lj + ε for a suitable integer γ. It would follow that L divides ε, contradicting 1 ≤ ε ≤ La/p < L. However, the fact that βp^{d+1−ℓ}/L is not an integer entails that

\frac{β}{L}p^{d+1−ℓ} − ⌊\frac{β}{L}p^{d+1−ℓ}⌋ ≥ \frac{1}{L},

as long as ℓ ≤ d + 1. Multiplication of both sides of this inequality by N_{k} leads to the chain of inequalities

\frac{N_{k}β}{L} p^{d+1−ℓ} − N_{k} ⌊\frac{β}{L}p^{d+1−ℓ}⌋ ≥ \frac{N_{k}}{L} ≥ 1

(it is here where we use the assumption L ≤ N_{k} = max(N_{1}, …, N_{k})), whence

⌊\frac{N_{k}β}{L} p^{d+1−ℓ}⌋ − N_{k} ⌊\frac{β}{L}p^{d+1−ℓ}⌋ ≥ 1,

provided ℓ ≤ d + 1. Use of this estimation in (5.8) gives

v_{p}(B_{N}(a + pj)) ≥ d + 1 = 1 + v_{p}(Lj + ε).

This completes the proof of (5.3), and, hence, of (5.1).

For later use, we record that we have in particular shown that for any

D ≤ 1 + max_{1≤ε≤⌊La/p⌋} v_{p}(Lj + ε)

we have

∑_{ℓ=2}^{D} (⌊\frac{N_{k}(a + pj)}{p^{ℓ}}⌋ − N_{k} ⌊\frac{a + pj}{p^{ℓ}} ⌋) ≥ D − 1.  (5.9)

We now embark on the proof of (3.5) itself.

5.2. Second part: the case j = 0. In this case, we want to prove that

B_{N}(a)H_{⌊La/p⌋} ∈ p\frac{M_{N}}{Θ_{L}} ℤ_{p} ,  (5.10)

or, using (3.3) (in the other direction), equivalently

B_{N}(a)H_{La} ∈ \frac{M_{N}}{Θ_{L}} ℤ_{p} ,  (5.11)

The reader should keep in mind that we still assume that p/L ≤ a < p, so that, in particular, a > 0.

If p > N_{k} = max(N_{1}, …, N_{k}), then our claim, in the form (5.10), reduces to B_{N}(a)H_{⌊La/p⌋} ∈ pℤ_{p}, which is indeed true because of (5.1) with j = 0.

Now let p ≤ N_{k}. Evidently, our claim, this time in the form (5.11), holds for a = 1. So, let a ≥ 2 from now on.

In a similar way as we did for the expression in (5.1), we bound the p-adic valuation of the expression in (5.11) from below:

v_{p}(B_{N}(a)H_{La}) = ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} (⌊\frac{N_{i}a}{p^{ℓ}}⌋ − N_{i} ⌊\frac{a}{p^{ℓ}} ⌋) + v_{p}(H_{La})

≥ ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} ⌊\frac{N_{i}a}{p^{ℓ}}⌋ − ⌊log_{p} La⌋

≥ ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} ⌊\frac{2N_{i}}{p^{ℓ}} ⌋ − ⌊log_{p} Lp⌋

≥ ⌊\frac{2N_{k}}{p} ⌋ + ∑_{ℓ=2}^{∞} ⌊\frac{2N_{k}}{p^{ℓ}} ⌋ + ∑_{i=1}^{k−1} ∑_{ℓ=1}^{∞} ⌊\frac{2N_{i}}{p^{ℓ}} ⌋ − ⌊log_{p} L⌋ − 1

≥ ⌊\frac{N_{k}}{p} ⌋ + ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} ⌊\frac{N_{i}}{p^{ℓ}} ⌋ − ⌊log_{p} L⌋ − 1  (5.12)

≥ max {1, ⌊L/p⌋} + ∑_{i=1}^{k} v_{p}(N_{i}!) − ⌊log_{p} L⌋ − 1.  (5.13)

If p = 2, then we can continue the estimation (5.13) as

v_{2}(B_{N}(a)H_{La}) ≥ ∑_{i=1}^{k} v_{2}(N_{i}!) − ⌊log_{2} L⌋ = v_{2}(M_{N}/Θ_{L}),  (5.14)

where we used Lemma 12 to obtain the equality. (In fact, at this point it was not necessary to consider the case p = 2 because a < p and because we assumed a ≥ 2. However, we shall re-use the present estimations later in the third part of the current proof, in a context where a = 1 is allowed.)

From now on let p ≥ 3. We use the fact that

k ≥ ⌊log_{p} k⌋ + 2  (5.15)

for all integers k ≥ 2 and primes p ≥ 3. Thus, in the case that L ≥ 2p, the estimation (5.13) can be continued as

v_{p}(B_{N}(a)H_{La}) ≥ 1 + ⌊log_{p} ⌊L/p⌋⌋ + ∑_{i=1}^{k} v_{p}(N_{i}!) − ⌊log_{p} L⌋ ≥ ∑_{i=1}^{k} v_{p}(N_{i}!) = v_{p}(M_{N}),

implying (5.14) in this case. If p ≤ L < 2p, then the estimation (5.13) can be continued as

v_{p}(B_{N}(a)H_{La}) ≥ 1 + ∑_{i=1}^{k} v_{p}(N_{i}!) − 2 = v_{p}(M_{N}/Θ_{L}),

implying (5.14) in this case also. Finally, if L < p, it follows from (5.13) that

v_{p}(B_{N}(a)H_{La}) ≥ 1 + ∑_{i=1}^{k} v_{p}(N_{i}!) − 1 = v_{p}(M_{N}),

implying (5.14) also in this final case. Everything combined, (5.11) follows immediately.

5.3. Third part: the case j > 0. Now let j > 0. If p > N_{k} = max(N_{1}, …, N_{k}), then (3.5) reduces to

B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) ∈ pℤ_{p} ,

which is again true because of (5.1).

Now let p ≤ N_{k}. The reader should keep in mind that we still assume that p/L ≤ a < p, so that, in particular, a > 0. In a similar way as we did for the expression in (5.1), we bound the p-adic valuation of the expression in (3.5) from below. For the sake of convenience, we write T_{1} for max_{1≤ε≤⌊La/p⌋} v_{p}(Lj + ε) and T_{2} for ⌊log_{p}(a + pj)⌋. Since it is somewhat hidden where our assumption j > 0 enters the subsequent considerations, we point out to the reader that j > 0 implies that T_{2} ≥ 1; without this property the split of the sum over ℓ into subsums in the chain of inequalities below would be impossible. So, using the above notation, we have (the detailed explanations for the various steps are given immediately after the following chain of estimations)

v_{p}(B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) )

= ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} (⌊\frac{N_{i}(a + pj)}{p^{ℓ}}⌋ − N_{i} ⌊\frac{a + pj}{p^{ℓ}} ⌋) + v_{p}(H_{Lj+⌊La/p⌋} − H_{Lj})

= ⌊\frac{N_{k}(a + pj)}{p}⌋ − N_{k} ⌊\frac{a + pj}{p} ⌋ + ∑_{ℓ=2}^{min\{1+T_{1},T_{2}\}} (⌊\frac{N_{k}(a + pj)}{p^{ℓ}}⌋ − N_{k} ⌊\frac{a + pj}{p^{ℓ}} ⌋)

+ ∑_{ℓ=min\{1+T_{1},T_{2}\}+1}^{∞} (⌊\frac{N_{k}(a + pj)}{p^{ℓ}}⌋ − N_{k} ⌊\frac{a + pj}{p^{ℓ}} ⌋)

+ ∑_{i=1}^{k−1} ∑_{ℓ=1}^{∞} (⌊\frac{N_{i}(a + pj)}{p^{ℓ}}⌋ − N_{i} ⌊\frac{a + pj}{p^{ℓ}} ⌋) + v_{p}(H_{Lj+⌊La/p⌋} − H_{Lj})

≥ ⌊\frac{N_{k}a}{p}⌋ + min\{1 + T_{1}, T_{2}\} − 1 + ∑_{i=1}^{k} ∑_{ℓ=T_{2}+1}^{∞} (⌊\frac{N_{i}(a + pj)}{p^{ℓ}}⌋ − N_{i} ⌊\frac{a + pj}{p^{ℓ}} ⌋)

+ v_{p}(H_{Lj+⌊La/p⌋} − H_{Lj})  (5.16)

≥ ⌊\frac{N_{k}a}{p}⌋ + T_{1} + v_{p}(H_{Lj+⌊La/p⌋} − H_{Lj}) + min\{0, T_{2} − T_{1} − 1\}

+ ∑_{i=1}^{k} ∑_{ℓ=⌊log_{p}(a+pj)⌋+1}^{∞} (⌊\frac{N_{i}(a + pj)}{p^{ℓ}}⌋ − N_{i} ⌊\frac{a + pj}{p^{ℓ}} ⌋)  (5.17)

≥ max \{1, ⌊L/p⌋\} + min\{0, T_{2} − T_{1} − 1\} + ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} ⌊\frac{N_{i}}{p^{ℓ}} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋  (5.18)

≥ max {1, ⌊L/p⌋} + ⌊log_{p}(a + pj)⌋ − ⌊log_{p} (Lj + ⌊La/p⌋)⌋ − 1

+ ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} ⌊\frac{N_{i}}{p^{ℓ}} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋  (5.19)

≥ max \{1, ⌊L/p⌋\} + ⌊log_{p} j⌋ − ⌊log_{p} (Lj + ⌊La/p⌋)⌋ + ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} ⌊\frac{N_{i}}{p^{ℓ}} ⌋  (5.20)

≥ max \{1, ⌊L/p⌋\} + ⌊log_{p} j⌋ − ⌊log_{p} L⌋ − ⌊log_{p} (j + \frac{1}{L}⌊La/p⌋)⌋ − 1

+ ∑_{i=1}^{k} v_{p}(N_{i}!)  (5.21)
≥ max {1, ⌊L/p⌋} − ⌊log_{p} L⌋ − 1 + v_{p}(M_{N}).  (5.22)

Here, we used (5.9) in order to get (5.16). To get (5.18), we used the inequalities

⌊\frac{N_{k}a}{p}⌋ ≥ ⌊\frac{N_{k}}{p} ⌋ ≥ max \{1, ⌊L/p⌋\}  (5.23)

and

T_{1} + v_{p}(H_{Lj+⌊La/p⌋} − H_{Lj}) ≥ 0.  (5.24)

To get (5.19), we used that

T_{2} − T_{1} − 1 ≥ ⌊log_{p}(a + pj)⌋ − ⌊log_{p} (Lj + ⌊La/p⌋)⌋ − 1

and

⌊log_{p}(a + pj)⌋ − ⌊log_{p} (Lj + ⌊La/p⌋)⌋ − 1 = ⌊log_{p} j⌋ − ⌊log_{p} (Lj + ⌊La/p⌋)⌋ ≤ 0,

so that

min{0, T_{2} − T_{1} − 1} ≥ ⌊log_{p}(a + pj)⌋ − ⌊log_{p} (Lj + ⌊La/p⌋)⌋ − 1.  (5.25)

Next, to get (5.20), we used

⌊\frac{N_{i}}{p^{ℓ}} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋ ≥ ⌊\frac{N_{i}}{p^{ℓ}} ⌋ .  (5.26)

To get (5.21), we used

⌊log_{p} (Lj + ⌊La/p⌋)⌋ ≤ ⌊log_{p} L⌋ + ⌊log_{p} (j + \frac{1}{L}⌊La/p⌋)⌋ + 1.  (5.27)

Finally, we used \frac{1}{L}⌊La/p⌋ < 1 in order to get (5.22).

If we now repeat the arguments after (5.13), then we see that the above estimation implies

v_{p}(B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) ) ≥ v_{p}(M_{N}/Θ_{L}).  (5.28)

This almost proves (3.5), our lower bound on the p-adic valuation of the number in (3.5) is just by 1 too low.

In order to establish that (3.5) is indeed correct, we assume by contradiction that all the inequalities in the estimations leading to (5.22) and finally to (5.28) are in fact equalities. In particular, the estimations in (5.23) hold with equality only if a = 1 and, if L should be at least p, also ⌊N_{k}/p⌋ = ⌊L/p⌋. We shall henceforth assume both of these two conditions.

If we examine the arguments after (5.13) that led us from (5.22) to (5.28), then we see that they prove in fact

v_{p}(B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) ) ≥ 1 + v_{p}(M_{N}/Θ_{L})  (5.29)

except if:

Case 1: p = 2 and ⌊L/2⌋ = 1;

Case 2: p ≥ 3 and p ≤ L < 2p;

Case 3: p = 3 and ⌊L/3⌋ = 2;

Case 4: L < p.

In all other cases, there holds either strict inequality in (5.15) with k = ⌊L/p⌋, or one has v_{p}(Θ_{L}) ≥ 1 and is able to show

v_{p}(B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) ) ≥ v_{p}(M_{N}),

so that (5.29) is satisfied, as desired. We now show that (5.29) holds as well in Cases 1–4, thus completing the proof of (3.5).

Case 1. Let first p = 2 and L = 2. We then have

min{0, T_{2} − T_{1} − 1} = min{0, ⌊log_{2}(2j + 1)⌋ − v_{2}(2j + 1) − 1}
= min{0, ⌊log_{2}(2j + 1)⌋ − 1} = 0 > −1,

in contradiction to having equality in (5.25).

On the other hand, if p = 2 and L = 3 then, because of equality in the second estimation in (5.23), we must have N_{k} = 3. We have

H_{Lj+⌊La/p⌋} − H_{Lj} = H_{3j+1} − H_{3j} = \frac{1}{3j + 1}.

If there holds equality in (5.25), then Lj + ⌊La/p⌋ = 3j + 1 must be a power of 2, say 3j + 1 = 2^{e} or, equivalently, j = (2^{e} − 1)/3. It follows that

⌊\frac{N_{k}}{p} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋ = ⌊\frac{3}{2} · \frac{1 + 2j}{2^{⌊log_{2}(1+2j)⌋}} ⌋ = ⌊\frac{3}{2} · \frac{2^{e+1} + 1}{3 · 2^{e−1}}⌋ = 2 > 1 = ⌊\frac{3}{2}⌋ = ⌊\frac{N_{k}}{p} ⌋ ,

in contradiction to having equality in (5.26) with ℓ = 1.

Case 2. Our assumptions p ≥ 3 and p ≤ L < 2p imply

H_{Lj+⌊La/p⌋} − H_{Lj} = H_{Lj+1} − H_{Lj} = \frac{1}{Lj + 1}.

Arguing as in the previous case, in order to have equality in (5.25), we must have Lj + 1 = f · p^{e} for some positive integers e and f with 0 < f < p. Thus, j = (f · p^{e} − 1)/L and p < L. (If p = L then j would be non-integral.) It follows that

⌊\frac{N_{k}}{p} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋ = ⌊\frac{N_{k}}{p} · \frac{f · p^{e+1} + L − p}{L · p^{⌊log_{p}((f·p^{e+1}+L−p)/L)⌋}} ⌋ .  (5.30)

If f = 1, then we obtain from (5.30) that

⌊\frac{N_{k}}{p} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋ = ⌊\frac{N_{k}}{p} · \frac{p^{e+1} + L − p}{L · p^{e−1}}⌋ ≥ ⌊\frac{p^{e+1} + L − p}{p^{e}}⌋ > 1 = ⌊\frac{L}{p}⌋ = ⌊\frac{N_{k}}{p} ⌋ ,

in contradiction with having equality in (5.26) with ℓ = 1.

On the other hand, if f ≥ 2, then we obtain from (5.30) that

⌊\frac{N_{k}}{p} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋ ≥ ⌊\frac{f · p^{e+1} + L − p}{p^{e+1}}⌋ ≥ f > 1 = ⌊\frac{L}{p}⌋ = ⌊\frac{N_{k}}{p} ⌋ ,

again in contradiction with having equality in (5.26) with ℓ = 1.

Case 3. Our assumptions p = 3 and ⌊L/3⌋ = 2 imply

H_{Lj+⌊La/p⌋} − H_{Lj} = H_{Lj+2} − H_{Lj} = \frac{1}{Lj + 1} + \frac{1}{Lj + 2}.

Similar to the previous cases, in order to have equality in (5.25), we must have Lj+ε = f ·3^{e} for some positive integers ε, e, f with 0 < ε, f < 3. The arguments from Case 2 can now be repeated almost verbatim. We leave the details to the reader.

Case 4. If L < p, then p/L > 1 = a, a contradiction to the assumption that we made at the very beginning of this section.

This completes the proof of the lemma.

6. Proof of Lemma 6

We want to use Proposition 1 with A_{r}(m) = g_{r}(m) = B_{N}(m) for all r. Clearly, the proposition would imply that S(a, K, s, p, m) ∈ p^{s+1}B_{N}(m)ℤ_{p}, and, thus, the claim. So, we need to verify the conditions (i)–(iii) in the statement of the proposition.

Condition (i) is true since B_{N}(0) = 1. Condition (ii) holds by the definitions of A_{r}(m) and g_{r}(m). To check that Condition (iii) holds is more complicated. The proof will be decomposed in three steps.

6.1. First step. Straightforward computations imply that, for any fixed j ∈ {1, 2, …, k}, we have

\frac{B_{N_{j}}(v + up + np^{s+1})}{B_{N_{j}}(up + np^{s+1})} = \frac{(N_{j}v + N_{j}up + N_{j}np^{s+1})!}{(v + up + np^{s+1})!^{N_{j}}} · \frac{(up + np^{s+1})!^{N_{j}}}{(N_{j}up + N_{j}np^{s+1})!}

= \frac{(N_{j}v + N_{j}up + N_{j}np^{s+1})(N_{j}v − 1 + N_{j}up + N_{j}np^{s+1}) ⋯ (N_{j}up + 1 + N_{j}np^{s+1})}{(v + up + np^{s+1})^{N_{j}}(v − 1 + up + np^{s+1})^{N_{j}} ⋯ (1 + up + np^{s+1})^{N_{j}}}

= \frac{(N_{j}v + N_{j}up)(N_{j}v − 1 + N_{j}up) ⋯ (1 + N_{j}up) + O(p^{s+1})}{(v + up)^{N_{j}}(v − 1 + up)^{N_{j}} ⋯ (1 + up)^{N_{j}} + O(p^{s+1})},

where O(R) denotes an element of Rℤ_{p}. Now we claim that this implies

\frac{B_{N_{j}}(v + up + np^{s+1})}{B_{N_{j}}(up + np^{s+1})} = \frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} + O(p^{s+1}).  (6.1)

Indeed, if v = 0, then this holds trivially. If v > 0, then, together with the hypothesis v < p, we infer that (v + up)(v − 1 + up) ⋯ (1 + up) is not divisible by p, and thus we have

\frac{1}{((v + up)(v − 1 + up) ⋯ (1 + up))^{N_{j}} + O(p^{s+1})}

= \frac{1}{((v + up)(v − 1 + up)^{N_{j}} ⋯ (1 + up))^{N_{j}}} (1 + O(p^{s+1})).

Hence,

\frac{(N_{j}v + N_{j}up)(N_{j}v − 1 + N_{j}up) ⋯ (1 + N_{j}up) + O(p^{s+1})}{(v + up)^{N_{j}}(v − 1 + up)^{N_{j}} ⋯ (1 + up)^{N_{j}} + O(p^{s+1})}

= \frac{(N_{j}v + N_{j}up)(N_{j}v − 1 + N_{j}up) ⋯ (1 + N_{j}up)}{(v + up)^{N_{j}}(v − 1 + up)^{N_{j}} ⋯ (1 + up)^{N_{j}}}

+ \frac{O(p^{s+1})}{(v + up)^{N_{j}}(v − 1 + up)^{N_{j}} ⋯ (1 + up)^{N_{j}}} ,

which proves (6.1) because

\frac{1}{(v + up)(v − 1 + up) ⋯ (1 + up)} ∈ ℤ_{p}  (6.2)

and

\frac{(N_{j}v + N_{j}up)(N_{j}v − 1 + N_{j}up) ⋯ (1 + N_{j}up)}{(v + up)^{N_{j}}(v − 1 + up)^{N_{j}} ⋯ (1 + up)^{N_{j}}} = \frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)}.  (6.3)

Note that (6.2) and (6.3) also imply that B_{N_{j}}(v + up)/B_{N_{j}}(up) ∈ ℤ_{p}, a property that will be used below.

We deduce from (6.1) that

∏_{j=1}^{k} \frac{B_{N_{j}}(v + up + np^{s+1})}{B_{N_{j}}(up + np^{s+1})} = ∏_{j=1}^{k} (\frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} + O(p^{s+1})) .  (6.4)

By expanding the product on the right-hand side of (6.4) and using that

B_{N_{j}}(v + up)/B_{N_{j}}(up) ∈ ℤ_{p},

we obtain our first main equality:

\frac{B_{N}(v + up + np^{s+1})}{B_{N}(up + np^{s+1})} = \frac{B_{N}(v + up)}{B_{N}(up)} + O(p^{s+1}).  (6.5)

6.2. Second step. The properties of Γ_{p} imply that

\frac{B_{N_{j}}(up + np^{s+1})}{B_{N_{j}}(u + np^{s})} = (−1)^{N_{j}−1} \frac{Γ_{p}(1 + N_{j}up + N_{j}np^{s+1})}{Γ_{p}(1 + up + np^{s+1})^{N_{j}}}  (6.6)
= (−1)^{N_{j}−1} \frac{Γ_{p}(1 + N_{j}up) + O(p^{s+1})}{Γ_{p}(1 + up)^{N_{j}} + O(p^{s+1})}  (6.7)
= (−1)^{N_{j}−1} \frac{Γ_{p}(1 + N_{j}up)}{Γ_{p}(1 + up)^{N_{j}}} (1 + O(p^{s+1}))  (6.8)
= \frac{B_{N_{j}}(up)}{B_{N_{j}}(u)} (1 + O(p^{s+1})),  (6.9)

where (i) of Lemma 9 is used to see (6.6) and (6.9), and (ii) is used for (6.7). Equation (6.8) holds because Γ_{p}(1 + up) and Γ_{p}(1 + N_{j}up) are both not divisible by p. Taking the product over j = 1, 2, …, k, we obtain from (6.9) our second main equality:

\frac{B_{N}(up + np^{s+1})}{B_{N}(u + np^{s})} = \frac{B_{N}(up)}{B_{N}(u)} (1 + O(p^{s+1})).  (6.10)

6.3. Third step. We now multiply the right-hand and left-hand sides of the main equalities (6.5) and (6.10): we get after simplification

\frac{B_{N}(v + up + np^{s+1})}{B_{N}(u + np^{s})} = \frac{B_{N}(v + up)}{B_{N}(u)} (1 + O(p^{s+1})) + \frac{B_{N}(up)}{B_{N}(u)} O(p^{s+1}).

We can rewrite this as

\frac{B_{N}(v + up + np^{s+1})}{B_{N}(v + up)} = \frac{B_{N}(u + np^{s})}{B_{N}(u)} (1 + O(p^{s+1})) + \frac{B_{N}(up)}{B_{N}(u)} · \frac{B_{N}(u + np^{s})}{B_{N}(v + up)} O(p^{s+1})

= \frac{B_{N}(u + np^{s})}{B_{N}(u)} + \frac{B_{N}(u + np^{s})}{B_{N}(u)} O(p^{s+1}) + \frac{B_{N}(u + np^{s})}{B_{N}(v + up)} O(p^{s+1}), (6.11)

where the last line holds because v_{p}(B_{N}(up)/B_{N}(u)) = 0.

If we compare (2.2) (with A_{r}(m) = g_{r}(m) = B_{N}(m)) and (6.11), we see that it only remains to prove that we have

\frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ \frac{B_{N}(n)}{B_{N}(v + up)} ℤ_{p} and \frac{B_{N}(u + np^{s})}{B_{N}(v + up)} ∈ \frac{B_{N}(n)}{B_{N}(v + up)} ℤ_{p}.  (6.12)

The first membership relation follows from Lemma 8 because the latter implies the stronger property

\frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ ( ∏_{j=1}^{k} B_{N_{j}}(n)) ℤ_{p} = B_{N}(n) ℤ_{p}.  (6.13)

For the second membership relation, we note that

\frac{B_{N}(u + np^{s})}{B_{N}(v + up)} = B_{N}(u) · \frac{B_{N}(u + np^{s})}{B_{N}(u)} · \frac{1}{B_{N}(v + up)}

∈ B_{N}(u) \frac{B_{N}(n)}{B_{N}(v + up)} ℤ_{p} ⊂ \frac{B_{N}(n)}{B_{N}(v + up)} ℤ_{p},

where we have used (6.13).

It now follows from (6.11) and (6.12) that we have

\frac{B_{N}(v + up + np^{s+1})}{B_{N}(v + up)} − \frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ \frac{B_{N}(n)}{B_{N}(v + up)} ℤ_{p},

i.e., we have checked that hypothesis (iii) in Proposition 1 holds in our situation. We can therefore apply this proposition and obtain exactly the statement of the lemma.

7. Proof of Lemma 7

The claim is trivially true if p divides m. We may therefore assume that p does not divide m for the rest of the proof. Let us write m = a + pj, with 0 < a < p. Then comparison with (3.5) shows that we are in a very similar situation here. Indeed, we may derive (3.10) from Lemma 5. In order to see this, we observe that

H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}} = ∑_{ε=1}^{Lap^{s}} \frac{1}{Ljp^{s+1} + ε}

= ∑_{ε=1}^{⌊La/p⌋} \frac{1}{Ljp^{s+1} + εp^{s+1}} +_{∑_{ε=1}^{Lap^{s}} ; p^{s+1}∤ε} \frac{1}{Ljp^{s+1} + ε}

= \frac{1}{p^{s+1}} (H_{Lj+⌊La/p⌋} − H_{Lj}) +_{∑_{ε=1}^{Lap^{s}} ; p^{s+1}∤ε} \frac{1}{Ljp^{s+1} + ε}.

Because of v_{p}(x + y) ≥ min{v_{p}(x), v_{p}(y)}, this implies

v_{p}(H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}}) ≥ min{−1 − s + v_{p}(H_{Lj+⌊La/p⌋} − H_{Lj}), −s}.

It follows that

v_{p}(B_{N}(m)(H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}}))

≥ −1 − s + min {v_{p}(B_{N}(a + pj)(H_{Lj+⌊La/p⌋} − H_{Lj})), 1 + v_{p}(B_{N}(a + pj))} .

Use of Lemmas 4 and 5 then completes the proof.

8. Outline of the proof of Theorem 3

In this section, we provide a brief outline of the proof of Theorem 3, reducing it to several lemmas and their corollaries. These are subsequently proved in the next section.

We must slightly “upgrade” the proof of Theorem 2 in the special case that N = (N, N, …, N) (with k occurrences of N) and L = N. To be precise, for all primes p, and for all non-negative integers K, a, and j with 0 ≤ a < p, we must prove

C(a + Kp) = ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{N(K−j)} − pH_{Na+Njp}) ∈ pΞ_{N}N!^{k}ℤ_{p}, (8.1)

where B_{N}(m) = \frac{(Nm)!^{k}}{m!^{kN}}, instead of “just” (3.2) for the present choice of parameters. We recall that, in view of a comparison of (1.6) with the definition of Ξ_{N}, there is only something to prove if v_{p}(H_{N}) > 0. Hence, Lemma 12 tells that there is nothing to prove if p = 2, and Lemma 13 together with Remarks 2(a) in the Introduction tells that, if p = 3, only the case N = 22 need to be considered. Therefore, in the remainder of this section, we always assume that p is a prime with 3 ≤ p ≤ N and v_{p}(H_{N}) > 0, and if p = 3 then N = 22.

There are two cases which can be treated directly: if K = a = 0, then C(0) = 0, and thus (8.1) holds trivially, whereas if K = 0 and a = 1, then C(1) = −pB_{N}(1)H_{N} = −pN!^{k}H_{N} and thus (8.1) holds by definition of Ξ_{N}. We therefore assume in addition a + Kp ≥ 2 for the remainder of this section.

Going through the outline of the proof of Theorem 2 in Section 3, we see that, in order to establish (8.1), we need to prove corresponding stronger versions of (3.4) and Lemmas 5 and 7. To be precise, given non-negative integers m, K, a, and j with 0 ≤ a < p and a + Kp ≥ 2, for p ≥ 5 we should prove

C(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{N(K−j)} − H_{⌊Na/p⌋+Nj}) mod p^{3}N!^{k}ℤ_{p}, (8.2)

respectively, for p = 3 and N = 22,

C(a + 3K) ≡ ∑_{j=0}^{K} B_{N}(a + 3j)B_{N}(K − j)(H_{N(K−j)} − H_{⌊Na/3⌋+Nj}) mod 3^{2}N!^{k}ℤ_{3}, (8.3)

respectively, if v_{p}(H_{N}) ≥ 3 and p a Wolstenholme prime or p | N,

C(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{N(K−j)} − H_{⌊Na/p⌋+Nj}) mod p^{4}N!^{k}ℤ_{p}, (8.4)

and we need to prove

B_{N}(a + pj) (H_{Nj+⌊Na/p⌋} − H_{Nj}) ∈ pΞ_{N}N!^{k}ℤ_{p}  (8.5)

and

B_{N}(m)(H_{Nmp^{s}} − H_{N⌊m/p⌋p^{s+1}}) ∈ p^{−s}Ξ_{N}N!^{k}ℤ_{p} .  (8.6)

If these five relations are used in Section 3 instead of their weaker counterparts (3.4), (3.5), and (3.10), respectively, then the proof in Section 3 yields (8.1), as required.

8.1. Proof of (8.2)–(8.4). We recall that the congruence (3.4) followed from the congruence (3.3). Clearly, one cannot hope for improving (3.3) to pH_{J} ≡ H_{⌊J/p⌋} mod p^{3}ℤ_{p}, there are counterexamples. However, if we combine Corollary 2 and Lemma 15(1) with L = N, then we see that

B_{N}(a + jp)B_{N}(K − j)pH_{Na+Njp} ≡ B_{N}(a + jp)B_{N}(K − j)H_{⌊Na/p⌋+Nj} mod p^{3}N!^{k}ℤ_{p}

as long as a + Kp ≥ 2 and a ≠ 0. Moreover, due to Lemma 4 and Corollary 3, the above congruence is even true if a = 0 and p ≥ 5. This implies (8.2). The congruence (8.3) follows in the same way by the slightly weaker assertion for p = 3 in Corollary 3.

For the congruence (8.4), one needs to combine Corollary 2 and Lemma 15(4) with L = N, to see that

B_{N}(a+jp)B_{N}(K −j)pH_{Na+Njp} ≡ B_{N}(a+jp)B_{N}(K −j)H_{⌊Na/p⌋+Nj} mod p^{4}N!^{k}ℤ_{p} (8.7)

as long as a + Kp ≥ 2 and a ≠ 0. If a = 0, then the congruence (8.7) still holds as long as j < K because of Lemma 4, Corollary 3, and the fact that the term B_{N}(K −j) contributes at least one factor p. The only remaining case to be discussed is a = 0 and j = K. If we apply the simple observation that v_{p}(B_{N}(p^{e}h)) = v_{p}(B_{N}(h)) for any positive integers e and h to B_{N}(a + jp) = B_{N}(jp), then, making again appeal to Corollary 2 and Lemma 15(4) with L = N, we see that (8.7) holds as well as long as j = K is no prime power. Finally, let a = 0 and j = K be a prime power, j = K = p^{e} say. If e ≥ 1, then we may use Lemma 17 with J = a + jp = a + Kp = p^{e+1} and v_{p}(B_{N}(p^{e+1})) = v_{p}(B_{N}(1)) = v_{p}(N!^{k}) to conclude that (8.7) also holds in this case. On the other hand, if j = K = 1, then Lemma 18, the assumption that p is a Wolstenholme prime or that p divides N, the fact that v_{p}(B_{N}(p)) = v_{p}(B_{N}(1)) = v_{p}(N!^{k}), altogether yield the congruence (8.7) in this case as well.

8.2. Proof of (8.5). The proof of Lemma 5 in Section 5 can be used verbatim. The place where the necessary improvement is possible is (5.15) (which was used there with k = ⌊log_{p} L⌋). Clearly, we have ⌊22/3⌋ ≥ ⌊log_{3} ⌊22/3⌋⌋ + 5. Moreover, we claim that in our more special context we have

⌊N/5⌋ ≥ ⌊log_{5} ⌊N/5⌋⌋ + 4  (8.8)

if p = 5, we have

⌊N/p⌋ ≥ ⌊log_{p} ⌊N/p⌋⌋ + 5  (8.9)

if p > 5, and we have

⌊N/p⌋ ≥ ⌊log_{p} ⌊N/p⌋⌋ + 6  (8.10)

if v_{p}(H_{N}) > 2. Indeed, if p = 5 then Lemma 14 says that N must be at least 20 because otherwise v_{5}(H_{N}) ≤ 0. The inequality (8.8) follows immediately. On the other hand, if p = 11 then it can be checked (by our table mentioned in Remarks 2(c) in the Introduction, for example) that N must be at least 77 because otherwise v_{11}(H_{N}) ≤ 0. If p is different from 5 and 11 (in addition to being at least 5) then, according to Lemma 15(3), we must have N/p ≥ 5. In both of the latter cases, the inequality (8.9) follows immediately. Finally, if v_{p}(H_{N}) > 2 then, according to Lemmas 14 and 15(4), we must have p > 5 and N/p ≥ 6, thus implying (8.10).

The effect of the above improvement over (5.15) is that the proof in Section 5 shows that in our more special context we have

B_{N}(a + pj) (H_{Nj+⌊Na/p⌋} − H_{Nj}) ∈ \begin{cases}p^{2}N!^{k}ℤ_{p} & if p = 5, \\ p^{4}N!^{k}ℤ_{p} & if v_{p}(H_{N}) > 2, \\ p^{3}N!^{k}ℤ_{p} & otherwise. \end{cases}

This implies (8.5) since v_{5}(H_{N}) ≤ 1 according to Lemma 14.

8.3. Proof of (8.6). Again, the proof of Lemma 7 in Section 7 can be used verbatim. The only differences arise at the very end. Namely, we have to use Corollary 2 combined with Lemma 15(1) instead of Lemma 4, and (8.2) instead of Lemma 5, to obtain our assertion for m ≥ 2. In the remaining case m = 1, we compute directly:

B_{N}(1)H_{Np^{s}} = N!^{k}(\frac{1}{p^{s}} H_{N} + \frac{1}{p^{s−1}}_{∑_{ε=1}^{Np} ; p∤ε} \frac{1}{ε} + \frac{1}{p^{s−2}}_{∑_{ε=1}^{Np^{2}} ; p∤ε} \frac{1}{ε} +_{∑_{ε=1}^{Np^{s}} ; p^{s−2}∤ε} \frac{1}{ε}).

The last sum over ε is clearly an element of p^{−s+3}ℤ_{p}. Moreover, if p ≥ 5, Lemma 16 implies that the next-to-last expression between parentheses is an element of p^{−s+4}ℤ_{p} ⊂ p^{−s+3}ℤ_{p}, and that the second expression between parentheses is an element of p^{−s+3}ℤ_{p}. If p = 3, it can be checked directly that the expression between parentheses is an element of 3^{−s+2}ℤ_{3}. Putting everything together, we conclude that

B_{N}(1)H_{Np^{s}} ∈ p^{−s}N!^{k}Ξ_{N}ℤ_{p},

which finishes the proof of (8.6).

9. More auxiliary lemmas

In this section, we prove the auxiliary results necessary for the proof of Theorem 3, of which the outline was given in the previous section. These are, on the one hand, improvements of Lemma 4 (see Lemmas 10, 11 and Corollary 2), and, on the other hand, assertions addressing specific p-adic properties of harmonic numbers (see Lemmas 12–18 and Corollary 3). Some of the results of this section are also referred to in the next section.

We begin by two lemmas improving on Lemma 4.

Lemma 10. For all positive integers N, a, and primes p with 2 ≤ a < p, we have

v_{p}(B_{N}(a)) ≥ ⌊\frac{N}{p} ⌋ + v_{p}(N!^{k}),

where N = (N, N, …, N) (with k occurrences of N).

Proof. This was implicitly proved by the estimations leading to (5.12) in the case that N_{i} = N for all i. □

Lemma 11. For all positive integers N, a, j, and primes p with 1 ≤ a < p, we have

v_{p}(B_{N}(a + jp)) ≥ ⌊\frac{N}{p} ⌋ + min\{1 + T_{1}, T_{2}\} − 1 + v_{p}(N!^{k}),

where N = (N, N, …, N) (with k occurrences of N), and where

T_{1} = max_{1≤ε≤⌊Na/p⌋} v_{p}(Nj + ε) and T_{2} = ⌊log_{p}(a + pj)⌋ .

Proof. This is seen by going through the estimations leading to (5.22) in the case that N_{i} = N for all i, without employing (5.23), (5.24), and (5.25). □

As a corollary to Lemmas 10 and 11, we obtain the following succinct p-adic estimation for B_{N}(m), which is needed in the proofs of (8.2), (8.4), and (8.6).

Corollary 2. Let N and m be positive integers and p be a prime such that m is at least 2 and not divisible by p. Then we have

v_{p}(B_{N}(m)) ≥ ⌊\frac{N}{p} ⌋ + v_{p}(N!^{k}),

where N = (N, N, …, N) (with k occurrences of N).

The next three lemmas of this section provide elementary information on the p-adic valuation of harmonic numbers for p = 2, 3, 5 which is needed in the proof of Lemma 15 and is also referred to frequently at other places. (For example, Lemma 12 was used in the proof of (5.14).) The proofs are not difficult (cf. [5]) and are therefore omitted.

Lemma 12. For all positive integers L, we have v_{2}(H_{L}) = − ⌊log_{2} L⌋.

Lemma 13. We have v_{3}(H_{2}) = v_{3}(H_{7}) = v_{3}(H_{22}) = 1. For positive integers L ∉ {2, 7, 22}, we have v_{3}(H_{L}) ≤ 0.

Lemma 14. We have v_{5}(H_{4}) = 2 and v_{5}(H_{20}) = v_{5}(H_{24}) = 1. For positive integers L ∉ {4, 20, 24}, we have v_{5}(H_{L}) ≤ 0.

Next, we record some properties of integers L and primes p for which v_{p}(H_{L}) > 0. These are needed throughout Section 8.

Lemma 15. Let p be a prime, and let L be an integer with p ≤ L. Then the following assertions hold true:

(1) If v_{p}(H_{L}) > 0 then L ≥ 2p.

(2) If v_{p}(H_{L}) > 0 and p ≠ 3 then L ≥ 3p.

(3) If v_{p}(H_{L}) > 0 and p ∉ {3, 5, 11} then L ≥ 5p.

(4) If v_{p}(H_{L}) > 2 then L ≥ 6p.

Proof. We have

H_{L} = \frac{1}{p}H_{⌊L/p⌋} +_{∑_{ε=1}^{L} ; p∤ε} \frac{1}{ε}.

Since the sum over ε is in ℤ_{p}, in order to have v_{p}(H_{L}) > 0 we must have v_{p}(H_{⌊L/p⌋}) > 0. Clearly, v_{p}(H_{1}) = 0 so that (1) follows. If ⌊L/p⌋ < 3 and p ≠ 3, then v_{p}(H_{⌊L/p⌋}) cannot be positive since H_{1} = 1 and H_{2} = \frac{3}{2}. This implies (2). If ⌊L/p⌋ < 5 and p ∉ \{3, 5, 11\}, then, again, v_{p}(H_{⌊L/p⌋}) cannot be positive since, as we already noted, H_{1} = 1 and H_{2} = \frac{3}{2}, and since H_{3} = \frac{11}{6} and H_{4} = \frac{25}{12}. This yields (3).

To see (4), we observe that, owing to Lemmas 12–14, we may assume that p ∉ {2, 3, 5}. Furthermore, if p = 11 then, according to our table referred to in Remarks 2(c) in the Introduction (see also [5]), we have L ≥ 848. Similarly, if p = 137 then, according to our table, we have L > 500000. The claim can now be established in the style of the proofs of (1)–(3) upon observing that H_{5} = \frac{137}{60}. □

We turn to a slight generalisation of Wolstenholme’s theorem on harmonic numbers. (We refer the reader to [13, Chapter VII] for information on this theorem, which corresponds to the case r = 1 in the lemma below.)

Lemma 16. For all primes p ≥ 5 and positive integers r, we have

v_{p}(H_{rp−1} − H_{rp−p}) ≥ 2.

Proof. By simple rearrangement, we have

H_{rp−1} − H_{rp−p} = ∑_{ε=1}^{p−1} \frac{1}{rp − p + ε} = ∑_{ε=1}^{(p−1)/2} (\frac{1}{rp − p + ε} + \frac{1}{rp − ε})

= p(2r − 1) ∑_{ε=1}^{(p−1)/2} \frac{1}{(rp − p + ε)(rp − ε)}.

It therefore suffices to consider the last sum over ε in ℤ/pℤ and show that it is ≡ 0 mod p. If we reduce this sum mod p, then we are left with

− ∑_{ε=1}^{(p−1)/2} \frac{1}{ε^{2}} ,

which is, up to the sign, the sum of all quadratic residues in ℤ/pℤ, that is, equivalently,

− ∑_{ε=1}^{(p−1)/2} ε^{2} = \frac{p(p − 1)(p + 1)}{24}.

Clearly, this is divisible by p for all primes p ≥ 5. □

As a corollary, we obtain strengthenings of (3.3) that we need in the proof of (8.2) and (8.3).

Corollary 3. For all primes p ≥ 5 and positive integers J divisible by p, we have

pH_{J} ≡ H_{J/p} mod p^{3}ℤ_{p}.

Moreover, for all positive integers J divisible by 3, we have

3H_{J} ≡ H_{J/3} mod 3^{2}ℤ_{3}.

Proof. By simple rearrangement, we have

pH_{J} − H_{J/p} = p ∑_{r=1}^{J/p}(H_{rp−1} − H_{rp−p}).

Due to Lemma 16, the p-adic valuation of this expression is at least 3 if p ≥ 5. If p = 3, it is easily seen directly that the 3-adic valuation of this expression is at least 2. □

Further strengthenings of (3.3), needed in the proof of (8.4), are given in the final two lemmas of this section.

Lemma 17. For all primes p ≥ 5 and positive integers J divisible by p^{2}, we have

pH_{J} ≡ H_{J/p} mod p^{5}ℤ_{p}.

Proof. Again, by simple rearrangement, we have

pH_{J} − H_{J/p} = p_{∑_{ε=1}^{J−1} ; p∤ε} \frac{1}{ε} = p ∑_{r=1}^{J/p^{2}} _{∑_{ε=1}^{p^{2}−1} ; p∤ε} \frac{1}{rp^{2} − p^{2} + ε}

= p^{3} ∑_{r=1}^{J/p^{2}}(2r − 1)_{∑_{ε=1}^{(p^{2}−1)/2} ; p∤ε} \frac{1}{(rp^{2} − p^{2} + ε)(rp^{2} − ε)}.

It therefore suffices to consider the last sum over ε in ℤ/p^{2}ℤ and show that it is ≡ 0 mod p^{2}. If we reduce this sum mod p^{2}, then we are left with

−_{∑_{ε=1}^{(p^{2}−1)/2} ; p∤ε} \frac{1}{ε^{2}} ,

which is, up to the sign, the sum of all quadratic residues in ℤ/p^{2}ℤ, that is, equivalently,

−_{∑_{ε=1}^{(p^{2}−1)/2} ; p∤ε}ε^{2} = − ∑_{ε=1}^{(p^{2}−1)/2} ε^{2} + ∑_{ε=1}^{(p−1)/2}(pε)^{2} = −\frac{p^{2}(p^{2} − 1)(p^{2} + 1)}{24} + p^{2} \frac{p(p − 1)(p + 1)}{24}.

Clearly, this is divisible by p^{2} for all primes p ≥ 5. □

Lemma 18. For all primes p ≥ 5 and positive integers N, we have

pH_{pN} ≡ H_{N} mod p^{4}ℤ_{p}  (9.1)

if and only if p is a Wolstenholme prime or p divides N.

Proof. Using a rearrangement in the spirit of Lemma 16, we obtain

pH_{pN} − H_{N} = p ∑_{r=1}^{N} ∑_{ε=1}^{p−1} \frac{1}{rp − p + ε}.

We consider the sum over r in ℤ/p^{3}ℤ. This leads to

∑_{r=1}^{N} ∑_{ε=1}^{p−1}(rp − p + ε)^{−1} ≡ ∑_{r=1}^{N} ∑_{ε=1}^{p−1} ε^{−1}(1 + p(r − 1)ε^{−1})^{−1}

≡ ∑_{r=1}^{N} ∑_{ε=1}^{p−1} (ε^{−1} − p(r − 1)ε^{−2} + p^{2}(r − 1)^{2}ε^{−3})

≡ NH_{p−1} − p\binom{N}{2}H_{p−1}^{(2)} + p^{2} \frac{N(N − 1)(2N − 1)}{6}H_{p−1}^{(3)} mod ℤ/p^{3}ℤ, (9.2)

where H_{m}^{(α)} denotes the higher harmonic number defined by H_{m}^{(α)} = ∑_{n=1}^{m} \frac{1}{n^{α}}. By a rearrangement analogous to the one in the proof of Lemma 16, one sees that v_{p}(H_{p−1}^{(3)}) ≥ 1, whence we may disregard the last term in the last line of (9.2). As it turns out, H_{p−1} and H_{p−1}^{(2)} are directly related modulo ℤ/p^{3}ℤ. Namely, we have

H_{p−1} ≡ ∑_{ε=1}^{p−1} ε^{−1} ≡ 2^{−1} ∑_{ε=1}^{p−1} (ε^{−1} + (p − ε)^{−1}) ≡ p2^{−1} ∑_{ε=1}^{p−1} (ε(p − ε))^{−1}

≡ −p2^{−1} ∑_{ε=1}^{p−1} ε^{−2}(1 + pε^{−1}) ≡ −p2^{−1} ∑_{ε=1}^{p−1} (ε^{−2} + pε^{−3})

≡ −p2^{−1}H_{p−1}^{(2)} − p^{2}2^{−1}H_{p−1}^{(3)} mod ℤ/p^{3}ℤ.  (9.3)

Again, we may disregard the last term. If we substitute this congruence in (9.2), then we obtain

∑_{r=1}^{N} ∑_{ε=1}^{p−1}(rp − p + ε)^{−1} ≡ N^{2}H_{p−1} mod ℤ/p^{3}ℤ.

Hence, the sum is congruent to zero mod ℤ/p^{3}ℤ if and only if N is divisible by p (recall Lemma 16 with r = 1) or v_{p}(H_{p−1}) ≥ 3, that is, if p is a Wolstenholme prime. This establishes the assertion of the lemma. □

10. The sequence (t_{N})_{N≥1}

The purpose of this section is to report on the evidence for our conjecture that the largest integer t_{N} such that q_{N,N}(z)^{1/t_{N}} ∈ ℤ[[z]] is given by t_{N} = Ξ_{N}N!, that is, that Theorem 3 with k = 1 is optimal. We assume k = 1 throughout this section.

We first prove that there cannot be any prime number p larger than N which divides t_{N}.

Proposition 2. Let p be a prime number and N a positive integer with p > N. Then there exists a positive integer a < p such that

v_{p}(B_{N}(a)H_{Na}) = 0,  (10.1)

where B_{N}(m) = \frac{(Nm)!}{m!^{N}}. In particular, p does not divide t_{N}.

Proof. If N = 1, we choose a = 1 to obtain B_{1}(1)H_{1} = 1. On the other hand, if N > 1, we choose a to be the least integer such that aN ≥ p. Since then a < p, we have

v_{p}(B_{N}(a)H_{Na}) = ∑_{ℓ=1}^{∞} (⌊\frac{Na}{p^{ℓ}}⌋ − N ⌊\frac{a}{p^{ℓ}} ⌋) + v_{p}(H_{Na}) = 1 − 1 = 0.

To see that p cannot divide t_{N}, we observe that we have C(a) = −pB_{N}(a)H_{Na} (for the sum C(·) defined in (3.1) in the case k = 1 and L = N) because a < p. Hence, the assertion (10.1) can be reformulated as v_{p}(C(a)) = 1. Since, because of p > N, we have v_{p}(M_{N}) = v_{p}(N!) = 0 and v_{p}(Ξ_{N}) = 0, it follows that

C(a) ∉ p^{2}Ξ_{N}N! ℤ_{p}.

This means that one cannot increase the exponent of p in (8.1) (with k = 1) in our special case, and thus p cannot divide t_{N}. □

So, if we hope to improve Theorem 3 with k = 1, then it must be by increasing exponents of prime numbers p ≤ N in (1.7). It can be checked directly that the exponent of 3 cannot be increased if N = 7. (The reader should recall Remarks 2(a) in the Introduction.) According to Remarks 2(b), an improvement is therefore only possible if v_{p}(H_{N}) > 2 for some p ≤ N. Lemmas 12–14 in Section 9 tell that this does not happen with p = 2, 3, 5, so that the exponents of 2, 3, 5 cannot be improved. (The same conclusion can also be drawn from [5] for many other prime numbers, but so far not for 83, for example.)

We already discussed in Remarks 2(c) whether at all there are primes p and integers N with p ≤ N and v_{p}(H_{N}) ≥ 3. We recall that, so far, only five examples are known, four of them involving p = 11.

The final result of this section shows that, if v_{p}(H_{N}) = 3, the exponent of p in the definition of Ξ_{N} in (1.7) cannot be increased so that Theorem 3 would still hold. (The reader should recall Remarks 2(b).)

Proposition 3. Let p be a prime number and N a positive integer with p ≤ N and v_{p}(H_{N}) = 3. If p is not a Wolstenholme prime and p does not divide N, then q_{N,N}(z)^{1/pΞ_{N} N!} ∉ ℤ[[z]].

Proof. We assume that p is not a Wolstenholme prime and that p does not divide N. By Lemmas 12–14, we can furthermore assume that p ≥ 7.

Going back to the outline of the proof of Theorem 3 in Section 8, we claim that

C(p) = (B_{N}(1)H_{N} − B_{N}(p)pH_{Np}) ∉ p^{4}N! ℤ_{p},  (10.2)

where B_{N}(m) = \frac{(Nm)!}{m!^{N}}. (The claim here is the non-membership relation; the equality holds by the definition of C(·) in (8.1).) This would imply that C(p) ∉ p^{2}Ξ_{N}N! ℤ_{p}, and thus, by Lemma 3 (recall the outlines of the proofs of Theorems 2 and 3 in Sections 3 and 8, respectively), that q_{N,N}(z)^{1/pΞ_{N} N!} ∉ ℤ[[z]], as desired.

To establish (10.2), we consider

H_{N}(B_{N}(1) − B_{N}(p))

= H_{N}N!(1 − \frac{1}{(p − 1)!^{N}} (1 · 2 ⋯ (p − 1))((p + 1) · (p + 2) ⋯ (2p − 1)) ⋯

× ⋯ ((pN − p + 1) · (pN − p + 2) ⋯ (pN − 1))).

Using v_{p}(H_{N}) = 3 and Wilson’s theorem, we deduce

H_{N}(B_{N}(1) − B_{N}(p)) ∈ p^{4}N! ℤ_{p}.  (10.3)

However, by Lemma 18 and the fact that v_{p}(B_{N}(p)) = v_{p}(B_{N}(1)) = v_{p}(N!), we obtain

B_{N}(1)H_{N} − B_{N}(p)pH_{pN} ≢ H_{N}(B_{N}(1) − B_{N}(p)) mod p^{4}N! ℤ_{p}.

Together with (10.3), this yields (10.2). □

To summarise the discussion of this section: if the conjecture in Remarks 2(c) that no prime p and integer N exists with v_{p}(H_{N}) ≥ 4 should be true, then Theorem 3 with k = 1 is sharp, that is, the sequence (t_{N})_{N≥1} is given by t_{N} = Ξ_{N}N!.

11. Sketch of the proof of Theorem 4

In this section we discuss the proof of Theorem 4. Since it is completely analogous to the proof of Theorem 3 (see Section 8), we content ourselves to point out the differences. At the end of the section, we present analogues of Propositions 2 and 3, addressing the question of optimality of Theorem 4 with k = 1.

First of all, by (1.4), we have

(z^{−1}q_{N}(z))^{1/kN} = exp(G̃_{N}(z)/F_{N}(z)),

where F_{N}(z) is the series from the Introduction and

G̃_{N}(z) := ∑_{m=1}^{∞} \frac{(Nm)!^{k}}{m!^{kN}} (H_{Nm} − H_{m}) z^{m}.

As in Section 8, we must “upgrade” the proof of Theorem 2 in the special case that N = (N, N, …, N). Writing as before B_{N}(m) = \frac{(Nm)!^{k}}{m!^{kN}}, we must consider the sum

C̃(a + Kp) := ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)((H_{N(K−j)} − H_{K−j}) − p(H_{Na+Njp} − H_{a+jp}))

= ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{N(K−j)} − pH_{Na+Njp})

− ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{K−j} − pH_{a+jp}) (11.1)

and show that it is in pΩ_{N}N!^{k}ℤ_{p} for all primes p, and for all non-negative integers K, a, and j with 0 ≤ a < p. The special cases K = a = 0, respectively K = 0 and a = 1, are equally simple to be handled directly here. We leave their verification to the reader and assume a + Kp ≥ 2 from now on.

Following the outline of the proof of Theorem 3 in Section 8, given non-negative integers m, K, a, and j with 0 ≤ a < p and a + Kp ≥ 2, we should prove

C̃(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)((H_{N(K−j)} − H_{K−j}) − (H_{⌊Na/p⌋+Nj} − H_{j}))

mod p^{3}N!^{k}ℤ_{p}, (11.2)

respectively, if v_{p}(H_{N} − 1) ≥ 3 and p a Wolstenholme prime or N ≡ ±1 mod p,

C̃(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)((H_{N(K−j)} − H_{K−j}) − (H_{⌊Na/p⌋+Nj} − H_{j}))

mod p^{4}N!^{k}ℤ_{p}, (11.3)

and we need to prove

B_{N}(a + pj) (H_{Nj+⌊Na/p⌋} − H_{Nj}) ∈ pΩ_{N}N!^{k}ℤ_{p}  (11.4)

and

B_{N}(m)((H_{Nmp^{s}} − H_{mp^{s}}) − (H_{N⌊m/p⌋p^{s+1}} − H_{⌊m/p⌋p^{s+1}})) ∈ p^{−s}Ω_{N}N!^{k}ℤ_{p} .  (11.5)

As the second line of (11.1) shows, C̃(a+Kp) is a difference of two sums C(a+Kp) as in (3.1), one with L = N, the other with L = 1. Hence, all the arguments in Section 8 which are not based on the assumption that v_{p}(H_{N}) > 0 can be used with only little modification.

On the other hand, the assumption v_{p}(H_{N}) > 0 is not used at many places. First, we need substitutes for Lemmas 12–15.

Lemma 19. For all positive integers L ≥ 2, we have v_{2}(H_{L} − 1) = − ⌊log_{2} L⌋.

Lemma 20. We have v_{3}(H_{66} − 1) = v_{3}(H_{68} − 1) = 1. For positive integers L ∉ {1, 66, 68}, we have v_{3}(H_{L} − 1) ≤ 0.

Lemma 21. We have v_{5}(H_{3} − 1) = v_{5}(H_{21} − 1) = v_{5}(H_{23} − 1) = 1. For positive integers L ∉ {1, 3, 21, 23}, we have v_{5}(H_{L} − 1) ≤ 0.

Lemma 22. Let p be a prime, and let L be an integer with L ≥ 2 and p ≤ L. Then the following assertions hold true:

(1) If v_{p}(H_{L} − 1) > 0 then L ≥ 4p.

(2) If v_{p}(H_{L} − 1) > 0 and p ≠ 5 then L ≥ 6p.

These results can be proved in exactly the same way as Lemmas 12–15, respectively. In comparison to Lemma 15, the statement of Lemma 22 is in fact simpler, so that complications that arose in Section 8 (such as (8.3), for example) do not arise here.

Second, we need a substitute for Lemma 18.

Lemma 23. For all primes p ≥ 5 and positive integers N, we have

p(H_{pN} − H_{p}) ≡ H_{N} − 1 mod p^{4}ℤ_{p}  (11.6)

if and only if p is a Wolstenholme prime or N ≡ ±1 mod p.

Proof. From the proof of Lemma 18, we know that

pH_{pN} − H_{N} ≡ pN^{2}H_{p−1} mod ℤ/p^{4}ℤ.

As a consequence, we obtain

p(H_{pN} − H_{p}) − (H_{N} − 1) ≡ p(N^{2} − 1)H_{p−1} mod ℤ/p^{4}ℤ.

The assertion of the lemma follows now immediately. □

Finally, the computation for m = 1 in Subsection 8.3 must be replaced by

B_{N}(1)(H_{Np^{s}} − H_{p^{s}}) = N!^{k}(\frac{1}{p^{s}} (H_{N} − 1) + \frac{1}{p^{s−1}}_{∑_{ε=p+1}^{Np} ; p∤ε} \frac{1}{ε} + \frac{1}{p^{s−2}}_{∑_{ε=p^{2}+1}^{Np^{2}} ; p∤ε} \frac{1}{ε} +_{∑_{ε=p^{s}+1}^{Np^{s}} ; p^{s−2}∤ε} \frac{1}{ε}),

with the conclusion that

B_{N}(1)(H_{Np^{s}} − H_{p^{s}}) ∈ p^{−s}N!^{k}Ω_{N}ℤ_{p}

being found in a completely analogous manner.

Altogether, this leads to a proof of Theorem 4.

12. The Dwork–Kontsevich sequence

In this section, we address the question of optimality of Theorem 4 when k = 1, that is, whether, given that k = 1, the largest integer u_{N} such that (z^{−1}q_{N}(z))^{\frac{1}{Nu_{N}}} ∈ ℤ[[z]] is given by Ω_{N}N!. Let us write q̃_{N}(z) for (z^{−1}q_{N}(z))^{1/N} with k being fixed to 1. The first proposition shows that there cannot be any prime number p larger than N which divides u_{N}. We omit the proof since it is completely analogous to the proof of Proposition 2 in Section 10.

Proposition 4. Let p be a prime number and N a positive integer with p > N ≥ 2. Then there exists a positive integer a < p such that

v_{p}(B_{N}(a)(H_{Na} − H_{a})) = 0.  (12.1)

In particular, p does not divide u_{N}.

So, if we hope to improve Theorem 4 with k = 1, then it must be by increasing exponents of prime numbers p ≤ N in (1.10). According to Remarks 3(b) in the Introduction, an improvement is therefore only possible if v_{p}(H_{N} − 1) > 2 for some p ≤ N.

The final result of this section shows that, if v_{p}(H_{N} − 1) = 3 (for which, however, so far no examples are known; see Remarks 3(b)), the exponent of p in the definition of Ω_{N} in (1.10) cannot be increased so that Theorem 4 with k = 1 would still hold.

Proposition 5. Let p be a prime number and N a positive integer with p ≤ N and v_{p}(H_{N}) = 3. If p is not a Wolstenholme prime and N ≢ ± 1 mod p, then q̃_{N}(z)^{\frac{1}{pΩ_{N} N!}} ∉ ℤ[[z]].

Again, the proof is completely analogous to the proof of Proposition 3 in Section 10, which we therefore omit.

So, if the conjecture in Remarks 3(c) that no prime p and integer N exists with v_{p}(H_{N} − 1) ≥ 4 should be true, then Theorem 4 with k = 1 is optimal, that is, the Dwork–Kontsevich sequence (u_{N})_{N≥1} is given by u_{N} = Ω_{N}N!.

13. Outline of the proof of Theorem 5

In this section, we provide a brief outline of the proof of Theorem 5, reducing it to Lemmas 24–26. These lemmas are subsequently proved in Sections 15 and 16, with two auxiliary lemmas being the subject of the subsequent section.

We follow the strategy that we used in Section 3 to prove Theorem 2; that is, by the consequence of Dwork’s Lemma given in Lemma 3, we want to prove that

F_{N}(z)G_{L,N}(z^{p}) − pF_{N}(z^{p})G_{L,N}(z) ∈ pzℤ_{p}[[z]].

The (a + Kp)-th Taylor coefficient of F_{N}(z)G_{L,N}(z^{p}) − pF_{N}(z^{p})G_{L,N}(z) is

C(a + Kp) := ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{L(K−j)} − pH_{La+Ljp}),

where B_{N}(m) := ∏_{j=1}^{k} B_{N_{j}}(m), the quantities B_{N_{j}}(m) being defined in (1.15). In view of Lemma 3, proving Theorem 5 is equivalent to proving that

C(a + Kp) ∈ pℤ_{p}

for all primes p and non-negative integers a and K with 0 ≤ a < p. Again using (3.3) with J = La + Ljp, we have

C(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{L(K−j)} − H_{⌊La/p⌋+Lj}) mod pℤ_{p}.

The analogue of Lemma 5 in the present context, which allows us to get rid of the floor function ⌊La/p⌋ and rearrange the sum over j, is the following lemma. The proof can be found in Section 15.

Lemma 24. For any prime p, non-negative integers a and j with 0 ≤ a < p, positive integers N_{1}, N_{2}, …, N_{k}, and L ∈ {1, 2, …, max(N_{1}, …, N_{k})}, we have

B_{N}(a + pj)(H_{Lj+⌊La/p⌋} − H_{Lj}) ∈ pℤ_{p}.

We now do the same rearrangements as those after Lemma 5 to conclude that

C(a + Kp) ≡ − ∑_{s=0}^{r} ∑_{m=0}^{p^{r+1−s}−1} Y_{m,s} mod pℤ_{p},

where r is such that K < p^{r}, and

Y_{m,s} := (H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}})S(a, K, s, p, m),

the expression S(a, K, s, p, m) being defined by

S(a, K, s, p, m) := ∑_{j=mp^{s}}^{(m+1)p^{s}−1} (B_{N}(a + jp)B_{N}(K − j) − B_{N}(j)B_{N}(a + (K − j)p)).

In this expression for S(a, K, s, p, m), it is assumed that B_{N}(n) = 0 for negative integers n. If we prove that

Y_{m,s} ∈ pℤ_{p}  (13.1)

for all m and s, then C(a + Kp) ∈ pℤ_{p}, as desired.

Now, the last assertion follows from the following two lemmas, with the proof of the first given in Section 16, while the proof of the second is easily accomplished by a trivial adaptation of the proof of Lemma 7 given in Section 7, where we change all occurrences of B_{N} by B_{N} and apply Lemma 24 instead of Lemma 5. The use of Lemma 4 can be dropped without substitute. (^{9}) Lemma 25 is the analogue of Lemma 6 in the present context, while Lemma 26 is the analogue of Lemma 7.

Lemma 25. For all primes p and non-negative integers a, m, s, K with 0 ≤ a < p, we have

S(a, K, s, p, m) ∈ p^{s+1}B_{N}(m)ℤ_{p}.

9To prove the refinement announced at the end of the Introduction that q_{1,N}(z)^{1/B_{N}(1)} ∈ ℤ[[z]], the use of Lemma 4 must be everywhere replaced by the use of Lemma 30, the latter being proved in Section 14.

Lemma 26. For all primes p, non-negative integers m, positive integers N_{1}, N_{2}, …, N_{k}, and L ∈ {1, 2, …, max(N_{1}, …, N_{k})}, we have

B_{N}(m) (H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}}) ∈ \frac{1}{p^{s}} ℤ_{p}.

It is clear that Lemmas 25 and 26 imply (13.1). This completes the outline of the proof of Theorem 5.

14. Further auxiliary lemmas

In this section, we establish four auxiliary results. The first one, Lemma 27, provides the observation (1.17) that reduces Zudilin’s Conjecture 2 to Theorem 5. The second one, Lemma 28, is required for the proof of Lemma 24 in Section 15, while the third one, Lemma 29, is required for the proof of Lemma 25 in Section 16. The last result, Lemma 30, justifies an assertion made in the Introduction. Moreover, the proofs of Lemmas 29 and 30 make themselves use of Lemma 28.

Lemma 27. Let m be a non-negative integer, and let N be a positive integer with associated parameters α_{i}, β_{i}, μ, η (that is, given by (1.13) and (1.14), respectively). Then

H_{N}(m) = ∑_{j=1}^{μ} α_{j}H_{α_{j}m} − ∑_{j=1}^{η} β_{j}H_{β_{j}m}.

Proof. For N = 1, we have H_{1}(m) = 0, so that the assertion of the lemma holds trivially. Therefore, from now on, we assume N ≥ 2.

We claim that, for any real number m ≥ 0, we have

\frac{C_{N}^{m}}{Γ(m + 1)^{φ(N)}} ∏_{j=1}^{φ(N)} \frac{Γ(m + r_{j}/N)}{Γ(r_{j}/N)} = \frac{∏_{j=1}^{μ} Γ(α_{j}m + 1)}{∏_{j=1}^{η} Γ(β_{j}m + 1)},  (14.1)

where Γ(x) denotes the gamma function. This generalises Zudilin’s identity (1.15) to real values of m. We essentially extend his proof to real m, using the well-known formula [3, p. 23, Theorem 1.5.2]

Γ(a) Γ (a + \frac{1}{n}) Γ (a + \frac{2}{n}) ⋯ Γ (a + \frac{n − 1}{n}) = n^{\frac{1}{2} −an}(2π)^{(n−1)/2}Γ(an),  (14.2)

valid for real numbers a and positive integers n such that aN is not an integer ≤ 0. Indeed, as in the Introduction, let p_{1}, p_{2}, …, p_{ℓ} denote the distinct prime factors of N. (It should be noted that there is at least one such prime factor due to our assumption N ≥ 2.) Furthermore, for a subset J of {1, 2, …, ℓ}, let p_{J} denote the product ∏_{j∈J} p_{j} of corresponding prime factors of N. (In the case that J = ∅, the empty product must be interpreted as 1.) Then, by the principle of inclusion-exclusion, we can rewrite the left-hand side of (14.1) in the form

\frac{C_{N}^{m}}{Γ(m + 1)^{φ(N)}} · \frac{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| even \end{matrix}} ∏_{i=1}^{N/p_{J}} Γ (m + \frac{ip_{J}}{N} )}{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| odd \end{matrix}} ∏_{i=1}^{N/p_{J}} Γ (m + \frac{ip_{J}}{N} )} · \frac{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| odd \end{matrix}} ∏_{i=1}^{N/p_{J}} Γ (\frac{ip_{J}}{N} )}{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| even \end{matrix}} ∏_{i=1}^{N/p_{J}} Γ (\frac{ip_{J}}{N} )}.

To each of the products over i, formula (14.2) can be applied. As a result, we obtain the expression

\frac{C_{N}^{m}}{Γ(m + 1)^{φ(N)}} · \frac{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| even \end{matrix}} (\frac{N}{p_{J}} )^{−(m+ \frac{p_{J}}{N})\frac{N}{p_{J}}} Γ (m\frac{N}{p_{J}} + 1)}{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| odd \end{matrix}} (\frac{N}{p_{J}} )^{−(m+ \frac{p_{J}}{N})\frac{N}{p_{J}}} Γ (m\frac{N}{p_{J}} + 1)} · \frac{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| odd \end{matrix}}Γ (1)}{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| even \end{matrix}}Γ (1)}

= \frac{C_{N}^{m}}{Γ(m + 1)^{φ(N)}} · \frac{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| even \end{matrix}} (\frac{N}{p_{J}} )^{−mN/p_{J}} Γ (m\frac{N}{p_{J}} + 1)}{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| odd \end{matrix}} (\frac{N}{p_{J}} )^{−mN/p_{J}} Γ (m\frac{N}{p_{J}} + 1)}, (14.3)

where the simplification in the exponent of N/p_{J} is due to the fact that there are as many subsets of even cardinality of a given non-empty set as there are subsets of odd cardinality. Since, again by inclusion-exclusion,

∑_{J⊆\{1,2,...,ℓ\}} ; |J| even \frac{N}{p_{J}} −_{∑_{J⊆\{1,2,...,ℓ\}} ; |J| odd} \frac{N}{p_{J}} = N ∏_{p|N} (1 − \frac{1}{p}) = φ(N),  (14.4)

we have

\frac{1}{Γ(m + 1)^{φ(N)}} · \frac{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| even \end{matrix}}Γ (m\frac{N}{p_{J}} + 1)}{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| odd \end{matrix}}Γ (m\frac{N}{p_{J}} + 1)} = \frac{∏_{j=1}^{μ} Γ(α_{j}m + 1)}{∏_{j=1}^{η} Γ(β_{j}m + 1)}

and

\frac{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| even \end{matrix}}N^{−mN/p_{J}}}{_{\begin{matrix}∏ & J⊆\{1,2,...,ℓ\} \\ |J| odd \end{matrix}}N^{−mN/p_{J}}} = N^{−mφ(N)}.

Finally, consider a fixed prime number dividing N, p_{j} say. Then, using again (14.4), we see that the exponent of p_{j} in the expression (14.3) is

−\frac{m}{p_{j}}_{∑_{J⊆\{1,2,...,ℓ\}} ; |J| odd, j∉J} \frac{N}{p_{J}} + \frac{m}{p_{j}}_{∑_{J⊆\{1,2,...,ℓ\}} ; |J| even, j∉J} \frac{N}{p_{J}} = \frac{mN}{p_{j}} _{∏_{p|N} ; p≠p_{j}} (1 − \frac{1}{p}) = \frac{m}{p_{j}} \frac{φ(N)}{1 − \frac{1}{p_{j}}} = \frac{mφ(N)}{p_{j} − 1}.

If all these observations are used in (14.3), we arrive at the right-hand side of (14.1).

Now, let us call b(m) the function defined by both sides of (14.1), and let ψ(x) = Γ′(x)/Γ(x) be the digamma function. We will use the well-known property (see [3, p. 13, Theorem 1.2.7]) that ψ(x+n)−ψ(x) = H(x, n) for real numbers x > 0 and integers n ≥ 0.

By taking the logarithmic derivative of the right hand side of (14.1), we have

\frac{b′(m)}{b(m)} = ∑_{j=1}^{μ} α_{j}ψ(α_{j}m + 1) − ∑_{j=1}^{η} β_{j}ψ(β_{j}m + 1)

= ∑_{j=1}^{μ} α_{j}(ψ(1) + H_{α_{j}m}) − ∑_{j=1}^{η} β_{j}(ψ(1) + H_{β_{j}m})

= ∑_{j=1}^{μ} α_{j}H_{α_{j}m} − ∑_{j=1}^{η} β_{j}H_{β_{j}m},  (14.5)

because ∑_{j=1}^{μ} α_{j} = ∑_{j=1}^{η} β_{j}. It also follows that b′(0)/b(0) = 0.

On the other hand, by taking the logarithmic derivative of the left hand side of (14.1), we also have

\frac{b′(m)}{b(m)} = log(C_{N}) + ∑_{j=1}^{φ(N)} ψ(m + r_{j}/N) − φ(N)ψ(m + 1).

Since b′(0)/b(0) = 0, we have log(C_{N}) = − ∑_{j=1}^{φ(N)} ψ(r_{j}/N) + φ(N)ψ(1) and therefore,

\frac{b′(m)}{b(m)} = ∑_{j=1}^{φ(N)} (ψ(m + r_{j}/N) − ψ(r_{j}/N)) − φ(N)(ψ(m + 1) − ψ(1))

= ∑_{j=1}^{φ(N)} H(r_{j}/N, m) − φ(N)H(1, m)

= H_{N}(m).  (14.6)

The lemma follows by equating the expressions (14.5) and (14.6) obtained for b′(m)/b(m). □

Lemma 28. For any integer N ≥ 1 with associated parameters α_{i}, β_{i}, μ, η, the function

Δ(x) := ∑_{i=1}^{μ} ⌊α_{i}x⌋ − ∑_{i=1}^{η} ⌊β_{i}x⌋

has the following properties:

(i) Δ is 1-periodic. In particular, Δ(n) = 0 for all integers n.

(ii) For all integers n, Δ is weakly increasing on intervals of the form [n, n + 1).

(iii) For all real numbers x, we have Δ(x) ≥ 0.

(iv) For all rational numbers r ≠ 0 whose denominator is an element of {1, 2, …, N}, we have Δ(r) ≥ 1.

Remark 5. Clearly, the function Δ is a step function. The proof below shows that, in fact, all the jumps of Δ at non-integral places have the value +1 and occur exactly at rational numbers of the form r/N, with r coprime to N.

Proof. Property (i) follows from the equality ∑_{i=1}^{μ} α_{i} = ∑_{i=1}^{η} β_{i} and the trivial fact that Δ(0) = 0.

We turn our attention to property (ii). For convenience of notation, let

N = p_{1}^{e_{1}}p_{2}^{e_{2}} ⋯ p_{ℓ}^{e_{ℓ}}

be the prime factorisation of N, where, as before, p_{1}, p_{2}, …, p_{ℓ} are the distinct prime factors of N, and where e_{1}, e_{2}, …, e_{ℓ} are positive integers.

As we already observed in the remark above, the function Δ is a step function. Moreover, jumps of Δ can only occur at values of x where some of the α_{i}x, 1 ≤ i ≤ μ, or some of the β_{j}x, 1 ≤ j ≤ η, (or both) are integers. At these values of x, the value of a (possible) jump is the difference between the number of i’s for which α_{i}x is integral and the number of j’s for which β_{j}x is integral. In symbols, the value of the jump is

#{i : 1 ≤ i ≤ μ and α_{i}x ∈ ℤ} − #{j : 1 ≤ j ≤ η and β_{j}x ∈ ℤ}.  (14.7)

Let X be the place of a jump, X not being an integer. Then we can write X as

X = \frac{Z}{p_{1}^{f_{1}}p_{2}^{f_{2}} ⋯ p_{ℓ}^{f_{ℓ}}} ,

where f_{1}, f_{2}, …, f_{ℓ} are non-negative integers, not all zero, and where Z is a non-zero integer relatively prime to p_{1}^{f_{1}}p_{2}^{f_{2}} ⋯ p_{ℓ}^{f_{ℓ}}. Given

α_{i} = p_{1}^{a_{1}}p_{2}^{a_{2}} ⋯ p_{ℓ}^{a_{ℓ}}

with e_{1} +e_{2} +⋯+e_{ℓ} −(a_{1} +a_{2} +⋯+a_{ℓ}) even and 0 ≤ e_{k} −a_{k} ≤ 1 for each k = 1, 2, …, ℓ, the number α_{i}X will be integral if and only if a_{k} ≥ f_{k} for all k ∈ {1, 2, …, ℓ}. Similarly, given

β_{j} = p_{1}^{b_{1}}p_{2}^{b_{2}} ⋯ p_{ℓ}^{b_{ℓ}}

with e_{1} + e_{2} + ⋯ + e_{ℓ} − (b_{1} + b_{2} + ⋯ + b_{ℓ}) odd and 0 ≤ e_{k} − b_{k} ≤ 1 for each k = 1, 2, …, ℓ, the number β_{j}X will be integral if and only if b_{k} ≥ f_{k} for all k ∈ {1, 2, …, ℓ}. We do not have to take into account the β_{j}’s which are 1, because 1 · X = X is not an integer by assumption. For the generating function of vectors (c_{1}, c_{2}, …, c_{ℓ}) with e_{k} ≥ c_{k} ≥ f_{k} and e_{k} − c_{k} ≤ 1, we have

∑_{c_{1}=max{e_{1}−1,f_{1}}}^{e_{1}} ⋯ ∑_{c_{ℓ}=max{e_{ℓ}−1,f_{ℓ}}}^{e_{ℓ}} z^{e_{1}+···+e_{ℓ}−(c_{1}+···+c_{ℓ})} = ∏_{k=1}^{ℓ} (1 + z · min{1, e_{k} − f_{k}}) .

We obtain the difference in (14.7) (with X in place of x) by putting z = −1 on the left-hand side of this relation. The product on the right-hand side tells us that this difference is 0 in case that e_{k} ≠ f_{k} for some k, while it is 1 otherwise. Thus, all the jumps of the function Δ at non-integral places have the value +1.

Property (iii) follows now easily from (i) and (ii).

In order to prove (iv), we observe that the first jump of Δ in the interval [0, 1) occurs at x = 1/N. Thus, Δ(x) ≥ 1 for all x in [1/N, 1). This implies in particular that Δ(r) ≥ 1 for all the above rational numbers r in the interval [1/N, 1). That the same assertion holds in fact for all the above rational numbers r (not necessarily restricted to [1/N, 1)) follows now from the 1-periodicity of the function Δ. □

Lemma 29. For any integers m, r, w ≥ 0 such that 0 ≤ w < p^{r}, we have

\frac{B_{N}(w + mp^{r})}{B_{N}(m)} ∈ ℤ_{p}.  (14.8)

Proof. We first show that we can assume that m is coprime to p. Indeed, let us write m = hp^{t} with gcd(h, p) = 1. We have to prove that

\frac{B_{N}(w + hp^{r+t})}{B_{N}(hp^{t})} ∈ ℤ_{p}.

Since v_{p}(B_{N}(hp^{t})/B_{N}(h)) = 0 (as can be easily seen from (1.15) and Legendre’s formula (4.1)), this amounts to prove that

\frac{B_{N}(w + hp^{r+t})}{B_{N}(h)} ∈ ℤ_{p},

which is the content of the lemma with r + t instead of r and h instead of m, with w < p^{r} < p^{r+t}.

Therefore, from now on, we assume that gcd(m, p) = 1 (however, this assumption will only be used after (14.11)). Since v_{p}(B_{N}(mp^{r})/B_{N}(m)) = 0, we have to prove that

v_{p} (\frac{B_{N}(w + mp^{r})}{B_{N}(mp^{r})}) ≥ 0

or, in an equivalent form, that

∑_{j=1}^{k} ∑_{ℓ=1}^{∞} (( ∑_{i=1}^{μ_{j}} ⌊\frac{α_{i,j}(w + mp^{r})}{p^{ℓ}}⌋ − ∑_{i=1}^{η_{j}} ⌊\frac{β_{i,j}(w + mp^{r})}{p^{ℓ}}⌋ )

− ( ∑_{i=1}^{μ_{j}} ⌊\frac{α_{i,j}mp^{r}}{p^{ℓ}} ⌋ − ∑_{i=1}^{η_{j}} ⌊\frac{β_{i,j}mp^{r}}{p^{ℓ}} ⌋ )) ≥ 0, (14.9)

where α_{i,j}, β_{i,j}, μ_{j}, η_{j} are the parameters associated to N_{j}.

If ℓ ≤ r, then for any j ∈ {1, 2, …, k},

∑_{i=1}^{μ_{j}} ⌊\frac{α_{i,j}mp^{r}}{p^{ℓ}} ⌋ − ∑_{i=1}^{η_{j}} ⌊\frac{β_{i,j}mp^{r}}{p^{ℓ}} ⌋ = mp^{r−ℓ} (∑_{i=1}^{μ_{j}} α_{i,j} − ∑_{i=1}^{η_{j}} β_{i,j}) = 0.

Moreover,

∑_{i=1}^{μ_{j}} ⌊\frac{α_{i,j}(w + mp^{r})}{p^{ℓ}}⌋ − ∑_{i=1}^{η_{j}} ⌊\frac{β_{i,j}(w + mp^{r})}{p^{ℓ}}⌋ ≥ 0

because of Lemma 28(iii) with N = N_{j}. It therefore suffices to show

∑_{j=1}^{k} ∑_{ℓ=r+1}^{∞} (( ∑_{i=1}^{μ_{j}} ⌊\frac{α_{i,j}(w + mp^{r})}{p^{ℓ}}⌋ − ∑_{i=1}^{η_{j}} ⌊\frac{β_{i,j}(w + mp^{r})}{p^{ℓ}}⌋ )

− ( ∑_{i=1}^{μ_{j}} ⌊\frac{α_{i,j}mp^{r}}{p^{ℓ}} ⌋ − ∑_{i=1}^{η_{j}} ⌊\frac{β_{i,j}mp^{r}}{p^{ℓ}} ⌋ )) ≥ 0, (14.10)

(The reader should note the difference to (14.9) occurring in the summation bounds for ℓ.) For ℓ > r, set x_{ℓ} = {mp^{r}/p^{ℓ}} and y_{ℓ} = {(w+mp^{r})/p^{ℓ}}. Using again ∑_{i=1}^{μ_{j}} α_{i,j} −∑_{i=1}^{η_{j}} β_{i,j} = 0, we see that the left-hand side of (14.10) is equal to

∑_{j=1}^{k} ∑_{ℓ=r+1}^{∞} (( ∑_{i=1}^{μ_{j}} ⌊α_{i,j}y_{ℓ}⌋ − ∑_{i=1}^{η_{j}} ⌊β_{i,j}y_{ℓ}⌋ ) − ( ∑_{i=1}^{μ_{j}} ⌊α_{i,j}x_{ℓ}⌋ − ∑_{i=1}^{η_{j}} ⌊β_{i,j}x_{ℓ}⌋ )). (14.11)

We now claim that x_{ℓ} ≤ y_{ℓ} for ℓ > r. To see this, we begin by the observation that, since m and p are coprime and ℓ > r, the rational number m/p^{ℓ−r} is not an integer. It follows that

x_{ℓ} + \frac{1}{p^{ℓ−r}} = \{\frac{m}{p^{ℓ−r}} \} + \frac{1}{p^{ℓ−r}} ≤ 1.

Hence, since w < p^{r}, we infer that

x_{ℓ} + \frac{w}{p^{ℓ}} < 1.

On the other hand, we have

y_{ℓ} = \{\frac{w}{p^{ℓ}} + ⌊\frac{m}{p^{ℓ−r}} ⌋ + x_{ℓ}\} = \{\frac{w}{p^{ℓ}} + x_{ℓ}\} = \frac{w}{p^{ℓ}} + x_{ℓ}.

Since w ≥ 0, we obtain indeed y_{ℓ} ≥ x_{ℓ}, as we claimed.

Using x_{ℓ} ≤ y_{ℓ} together with Lemma 28(ii), we see that, for ℓ > r and j = 1, 2, … k, we have

∑_{i=1}^{μ_{j}} ⌊α_{i,j}y_{ℓ}⌋ − ∑_{i=1}^{η_{j}} ⌊β_{i,j}y_{ℓ}⌋ ≥ ∑_{i=1}^{μ_{j}} ⌊α_{i,j}x_{ℓ}⌋ − ∑_{i=1}^{η_{j}} ⌊β_{i,j}x_{ℓ}⌋ ,

which shows that the expression in (14.11) is non-negative, thus establishing (14.10) and also (14.9). This finishes the proof of the lemma. □

We conclude this section with a result which was announced near the end of the Introduction. It is used nowhere else, but we give it for the sake of completeness. It is a generalisation of Lemma 4. By the same techniques used to prove Theorem 2, it enables one to prove that q_{1,N}(z)^{1/B_{N}(1)} ∈ ℤ[[z]] (see Footnote 9).

Lemma 30. For any vector N and any integer m ≥ 1, we have that B_{N}(1) divides B_{N}(m).

Proof. Obviously, it is sufficient to prove the assertion for k = 1 and N = (N). Let Δ be the function associated to N as defined in Lemma 28. We want to prove that, for any prime p, we have v_{p}(B_{N}(m)) ≥ v_{p}(B_{N}(1)). We can assume that m and p are coprime because v_{p}(B_{N}(mp^{t})) = v_{p}(B_{N}(m)) for any integers m, t ≥ 0.

Now, when gcd(m, p) = 1, we have that

v_{p}(B_{N}(m)) = ∑_{ℓ=1}^{∞} Δ(m/p^{ℓ}) = ∑_{ℓ=1}^{∞} Δ({m/p^{ℓ}})

≥ ∑_{ℓ=1}^{∞} Δ(1/p^{ℓ}) = v_{p}(B_{N}(1)).

Here, we used the 1-periodicity of Δ for the second equality. For the inequality, we used that {m/p^{ℓ}} ≥ 1/p^{ℓ} (because gcd(m, p) = 1 implies that m/p^{ℓ} is not an integer) and the the (partial) monotonicity of Δ described in Lemma 28(ii). □

15. Proof of Lemma 24

We follow the first part of the proof of Lemma 5. We write α_{i,m}, β_{i,m}, μ_{m}, and η_{m} for the parameters associated to N_{m}, m = 1, 2, …, k. We may assume that max(N_{1}, …, N_{k}) = N_{k}. Then, using again Lemma 28(iii),

v_{p}(B_{N}(a + pj)) = ∑_{m=1}^{k} ∑_{ℓ=1}^{∞} (∑_{i=1}^{μ_{m}} ⌊\frac{α_{i,m}(a + pj)}{p^{ℓ}}⌋ − ∑_{i=1}^{η_{m}} ⌊\frac{β_{i,m}(a + pj)}{p^{ℓ}}⌋)

≥ ∑_{ℓ=1}^{∞} (∑_{i=1}^{μ_{k}} ⌊\frac{α_{i,k}(a + pj)}{p^{ℓ}}⌋ − ∑_{i=1}^{η_{k}} ⌊\frac{β_{i,k}(a + pj)}{p^{ℓ}}⌋) = ∑_{ℓ=1}^{∞} Δ_{k}(\frac{a + jp}{p^{ℓ}})

with

Δ_{k}(x) := ∑_{i=1}^{μ_{k}} ⌊α_{i,k}x⌋ − ∑_{i=1}^{η_{k}} ⌊β_{i,k}x⌋ .

We want to prove that

v_{p}(B_{N}(a + pj)) ≥ 1 + v_{p}(Lj + ε)  (15.1)

for any integer ε such that 1 ≤ ε ≤ ⌊La/p⌋. We have

\frac{a + jp}{p^{ℓ}} = \frac{a − pε/L}{p^{ℓ}} + \frac{pj + pε/L}{p^{ℓ}}.

15.1. First step. We claim that

Δ_{k}(\frac{a + jp}{p^{ℓ}}) ≥ Δ_{k}(\frac{pj + pε/L}{p^{ℓ}}) .  (15.2)

To see this, we first observe that

Δ_{k}(\frac{a + jp}{p^{ℓ}}) = Δ_{k}(\frac{a − pε/L}{p^{ℓ}} + \frac{pj + pε/L}{p^{ℓ}}) = Δ_{k}(\frac{a − pε/L}{p^{ℓ}} + \{\frac{pj + pε/L}{p^{ℓ}}\})

because Δ_{k} is 1-periodic.

We now claim that

0 ≤ \frac{a − pε/L}{p^{ℓ}} + \{\frac{pj + pε/L}{p^{ℓ}}\} < 1.  (15.3)

Indeed, positivity is clear and we now concentrate on the upper bound. We write j = up^{ℓ−1} + v with 0 ≤ v < p^{ℓ−1}. Hence,

\{\frac{pj + pε/L}{p^{ℓ}}\} = \{u + \frac{pv + pε/L}{p^{ℓ}}\} = \{\frac{v}{p^{ℓ−1}} + \frac{pε/L}{p^{ℓ}}\} .

Since 0 ≤ ε ≤ ⌊La/p⌋ < L, we have 0 ≤ \frac{pε/L}{p^{ℓ}} < 1/p^{ℓ−1} and therefore

0 ≤ \frac{v}{p^{ℓ−1}} + \frac{pε/L}{p^{ℓ}} < \frac{v}{p^{ℓ−1}} + \frac{1}{p^{ℓ−1}} ≤ 1

(where the last inequality holds by definition of v), whence

\{\frac{pj + pε/L}{p^{ℓ}}\} = \frac{pv + pε/L}{p^{ℓ}}.

Therefore, we have

\frac{a − pε/L}{p^{ℓ}} + \{\frac{pj + pε/L}{p^{ℓ}}\} = \frac{a − pε/L}{p^{ℓ}} + \frac{pv + pε/L}{p^{ℓ}} = \frac{a}{p^{ℓ}} + \frac{v}{p^{ℓ−1}} .

Since \frac{v}{p^{ℓ−1}} < 1 and a < p, we necessarily have

\frac{a}{p^{ℓ}} + \frac{v}{p^{ℓ−1}} < 1,

as desired.

Since \frac{a−pε/L}{p^{ℓ}} ≥ 0, it follows from Lemma 28(i),(ii) (with Δ = Δ_{k}) and (15.3) that

Δ_{k}(\frac{a − pε/L}{p^{ℓ}} + \{\frac{pj + pε/L}{p^{ℓ}}\}) ≥ Δ_{k}(\{\frac{pj + pε/L}{p^{ℓ}}\}) = Δ_{k}(\frac{pj + pε/L}{p^{ℓ}}) .

Thus, we have proved the claim (15.2) made at the beginning of this step.

15.2. Second step. Let us write Lj + ε = βp^{d}, where d = v_{p}(Lj + ε), so that

\frac{pj + pε/L}{p^{ℓ}} = \frac{βp^{d+1−ℓ}}{L} .

We have proved in the first step that

v_{p}(B_{N}(a + pj)) ≥ ∑_{ℓ=1}^{∞} Δ_{k}(\frac{βp^{d+1−ℓ}}{L} ) .  (15.4)

By the same argument as the one that we used in the first part of the proof of Lemma 5 in Section 5, the rational number \frac{βp^{d+1−ℓ}}{L} is not an integer and for ℓ ≤ d + 1, the denominator of \frac{βp^{d+1−ℓ}}{L} is at most L. Since L ≤ N_{k}, it follows then from Lemma 28(iv), again with Δ = Δ_{k}, that Δ_{k}(βp^{d+1−ℓ}/L) ≥ 1 for any ℓ in {1, 2, …, d + 1}. Use of this estimation in (15.4) gives

v_{p}(B_{N}(a + pj)) ≥ d + 1 = 1 + v_{p}(Lj + ε).

This completes the proof of (15.1) and, hence, of Lemma 24.

16. Proof of Lemma 25

Again, we want to use Proposition 1, this time with A_{r}(m) = g_{r}(m) = B_{N}(m). Clearly, the proposition would imply that S(a, K, s, p, m) ∈ p^{s+1}B_{N}(m)ℤ_{p}, and, thus, the claim. So, we need to verify the conditions (i)–(iii) in the statement of the proposition.

Conditions (i) and (ii) of Dwork’s congruences theorem clearly hold, and we must check Condition (iii). To do the latter, we follow the method developed to prove Lemma 6 (see Section 6). The reader should recall that

B_{N}(m) := ∏_{j=1}^{k} B_{N_{j}}(m),

where B_{N_{j}}(m) is given by (1.12) or (1.15).

16.1. First step. Let us fix j ∈ {1, 2, …, k} and set D_{N} := N^{−φ(N)}C_{N}, which is an integer. We claim that

\frac{B_{N_{j}}(v + up + np^{s+1})}{B_{N_{j}}(up + np^{s+1})} = \frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} + O(p^{s+1}).  (16.1)

To prove (16.1), we can use the same arguments as in the first step of the proof of Lemma 6, due to the identities (^{10})

\frac{B_{N_{j}}(v + up + np^{s+1})}{B_{N_{j}}(up + np^{s+1})} = \frac{D_{N_{j}}^{v} ∏_{ℓ=1}^{φ(N_{j})} ∏_{i=1}^{v} (r_{ℓ} + (i − 1)N_{j} + uN_{j}p + nN_{j}p^{s+1})}{((v + up + np^{s+1})(v − 1 + up + np^{s+1}) ⋯ (1 + up + np^{s+1}))^{φ(N_{j})}} ,

\frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} = \frac{D_{N_{j}}^{v} ∏_{ℓ=1}^{φ(N_{j})} ∏_{i=1}^{v} (r_{ℓ} + (i − 1)N_{j} + uN_{j}p)}{((v + up)(v − 1 + up) ⋯ (1 + up))^{φ(N_{j})}},  (16.2)

and to the fact that (v + up)(v − 1 + up) ⋯ (1 + up) is not divisible by p.

A side result of (16.2) (which was actually used to prove (16.1)) is that

\frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} ∈ ℤ_{p}.

\frac{B_{N_{j}}(v + up + np^{s+1})}{B_{N_{j}}(up + np^{s+1})} = \frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} (1 + O(p^{s+1})).

10They are immediate consequences of the definition (1.12) of B_{N}. Zudilin used them in his proof of the following stronger version of (16.1):

However, for this, he assumes that p divides N_{j} (see [30, Eq. (35)]). Here, we do not assume that p divides N_{j}, and therefore we obtain the weaker equality (16.1), which is fortunately enough for our purposes.

We deduce from this fact and from (16.1) that

∏_{j=1}^{k} \frac{B_{N_{j}}(v + up + np^{s+1})}{B_{N_{j}}(up + np^{s+1})} = ∏_{j=1}^{k} (\frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} + O(p^{s+1})) = ∏_{j=1}^{k} \frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)} + O(p^{s+1}),

or, in other words,

\frac{B_{N}(v + up + np^{s+1})}{B_{N}(up + np^{s+1})} = \frac{B_{N}(v + up)}{B_{N}(up)} + O(p^{s+1}).  (16.3)

16.2. Second step. Let us fix j ∈ {1, 2, …, k}. Exactly as in the proof of the second step of Lemma 6, the properties of Γ_{p} imply that

\frac{B_{N_{j}}(up + np^{s+1})}{B_{N_{j}}(u + np^{s})} = (−1)^{μ_{j}−η_{j}} \frac{∏_{i=1}^{μ_{j}} Γ(1 + α_{i,j}(up + np^{s+1}))}{∏_{i=1}^{η_{j}} Γ(1 + β_{i,j}(up + np^{s+1}))}

= (−1)^{μ_{j}−η_{j}} \frac{∏_{i=1}^{μ_{j}} Γ(1 + α_{i,j}up) + O(p^{s+1})}{∏_{i=1}^{η_{j}} Γ(1 + β_{i,j}up) + O(p^{s+1})}

= (−1)^{μ_{j}−η_{j}} \frac{∏_{i=1}^{μ_{j}} Γ(1 + α_{i,j}up)}{∏_{i=1}^{η_{j}} Γ(1 + β_{i,j}up)} (1 + O(p^{s+1}))

= \frac{B_{N_{j}}(up)}{B_{N_{j}}(u)} (1 + O(p^{s+1})).

Here we used again the fact that Γ_{p}(m) is never divisible by p for any integer m. Hence, taking the product over j = 1, 2, …, k, we obtain

\frac{B_{N}(up + np^{s+1})}{B_{N}(u + np^{s})} = \frac{B_{N}(up)}{B_{N}(u)} (1 + O(p^{s+1})).  (16.4)

16.3. Third step. We follow verbatim the beginning of the third step in the proof of Lemma 6 (i.e., we multiply the left-hand and right-hand sides of (16.3) and (16.4), etc.) to obtain

\frac{B_{N}(v + up + np^{s+1})}{B_{N}(v + up)} − \frac{B_{N}(u + np^{s})}{B_{N}(u)} = \frac{B_{N}(u + np^{s})}{B_{N}(u)} O(p^{s+1}) + \frac{B_{N}(u + np^{s})}{B_{N}(v + up)} O(p^{s+1}).

It remains to check that

\frac{B_{N}(u + np^{s})}{B_{N}(u)} ∈ \frac{B_{N}(n)}{B_{N}(v + up)} ℤ_{p} and \frac{B_{N}(u + np^{s})}{B_{N}(v + up)} ∈ \frac{B_{N}(n)}{B_{N}(v + up)} ℤ_{p}.  (16.5)

The first assertion in (16.5) can be rewritten as

\frac{B_{N}(u + np^{s})}{B_{N}(n)} · \frac{B_{N}(v + up)}{B_{N}(u)} ∈ ℤ_{p},  (16.6)

while the second assertion can be rewritten as

\frac{B_{N}(u + np^{s})}{B_{N}(n)} ∈ ℤ_{p}.  (16.7)

Now, the assertion (16.7) is the special case w = u, m = n and r = s of Lemma 29, while (16.6) follows from (16.7) combined with the special case w = v, m = u and r = 1 of Lemma 29.

This completes the proof of the lemma.

Acknowledgements

The authors are extremely grateful to Alessio Corti and Catriona Maclean for illuminating discussions concerning the geometric side of our work, and to David Boyd for helpful information on the p-adic behaviour of the harmonic numbers H_{N} and for communicating to us the value (1.8) from his files from 1994.

Bibliography

[1] G. Almkvist and W. Zudilin, Differential equations, mirror maps and zeta values, in: Mirror Symmetry V, N. Yui, S.-T. Yau, and J.D. Lewis (eds.), AMS/IP Studies in Advanced Mathematics 38 (2007), International Press & Amer. Math. Soc., 481–515.

[2] Y. André, G-fonctions et transcendance, J. reine angew. Math. 476 (1996), 95–125.

[3] G. E. Andrews, R. A. Askey et R. Roy, Special Functions, The Encyclopedia of Mathematics and Its Applications, vol. 71, (G.-C. Rota, ed.), Cambridge University Press, Cambridge (1999).

[4] V. V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi– Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), 493–533.

[5] D. W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math. 3 (1994), 287–302.

[6] P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B359 (1991), 21–74.

[7] A. Corti and V. Golyshev, Hypergeometric equations and weighted projective spaces, preprint 2006;

   available at http://arχiv.org/abs/0607.5016.

[8] E. Cotterill, Rational curves of degree 10 on a general quintic threefold, Comm. Algebra 33 (2005), no. 6, 1833–1872.

[9] C. F. Doran, Picard–Fuchs uniformization and modularity of the mirror map, Comm. Math. Phys. 212 (2000), 625–647.

[10] B. Dwork, p-Adic cycles, Publ. I.H.E.S. 37 (1969), 27–115.

[11] B. Dwork, On p-adic differential equations IV: generalized hypergeometric functions as p-adic analytic functions in one variable, Ann. Sci. É.N.S. (4) 6, no. 3 (1973), 295–316.

[12] A. B. Givental, Equivariant Gromov–Witten invariants, Internat. Math. Res. Notices 13 (1996), 613– 663.

[13] G. H. Hardy et E. M. Wright, An introduction to the theory of number theory, 5th edition, Oxford University Press, 1979.

[14] N. Heninger, E. M. Rains and N. J. A. Sloane, On the integrality of nth roots of generating functions, J. Combin. Theory Ser. A 113 (2006), 1732–1745.

[15] S. Hosono, A. Klemm, S. Theisen, S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi–Yau spaces, Nuclear Phys. B 43, no. 3 (1995), 501–552.

[16] M. Kontsevich, A. Schwarz and V. Vologodsky, Integrality of instanton numbers and p-adic B-model, Phys. Lett. B 637 (2006), 97–101.

[17] S. Lang, Cyclotomic fields II, Graduate Texts in Mathematics 69, Springer–Verlag, 1980.

[18] B. H. Lian and S.-T. Yau, Arithmetic properties of mirror map and quantum coupling, Comm. Math. Phys. 176, no. 1 (1996), 163–191.

[19] B. H. Lian and S.-T. Yau, Mirror maps, modular relations and hypergeometric series I, appeared as Integrality of certain exponential series, in: Lectures in Algebra and Geometry, Proceedings of the International Conference on Algebra and Geometry, Taipei, 1995, M.-C. Kang (ed.), Int. Press, Cambridge, MA, 1998, pp. 215–227.

[20] B. H. Lian, K. Liu and S.-T. Yau, Mirror principle. I., Asian J. Math. 1, no. 4 (1997), 729–763.

[21] B. H. Lian and S.-T. Yau, The nth root of the mirror map, in: Calabi–Yau varieties and mirror symmetry, Proceedings of the Workshop on Arithmetic, Geometry and Physics around Calabi–Yau Varieties and Mirror Symmetry, Toronto, ON, 2001, N. Yui and J. D. Lewis (eds.), Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003, pp. 195–199.

[22] D. R. Morrison, Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993), 223–247.

[23] The On-line encyclopedia of integers sequences, http://www.research.att.com/~njas/sequences/.

[24] R. Pandharipande, Rational curves on hypersurfaces (after A. Givental), Séminaire Bourbaki. Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 848, 5, 307–340.

[25] J. Stienstra, GKZ Hypergeometric Structures, Proceedings of the Summer School “Algebraic Geometry and Hypergeometric Functions”, Istanbul, June 2005, preprint 2005; available at

   http://arχiv.org/abs/0511.5351.

[26] F. Rodriguez–Villegas, Hypergeometric families of Calabi–Yau manifolds, Proceedings of the Workshop on Arithmetic, Geometry and Physics around Calabi–Yau Varieties and Mirror Symmetry, Toronto, ON, 2001, N. Yui and J. D. Lewis (eds.), Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003, 223–231.

[27] C. Voisin, Mirror symmetry, SMF/AMS Texts and Monographs, vol. 1, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 1999.

[28] V. Vologodsky, Integrality of instanton numbers, preprint 2007; http://arχiv.org/abs/0707.4617.

[29] M. Yoshida, Fuchsian differential equations, Aspects of Mathematics 11, Vieweg, 1987.

[30] W. Zudilin, Integrality of power expansions related to hypergeometric series, Mathematical Notes 71.5 (2002), 604–616.

C. Krattenthaler, Fakultät für Mathematik, Universität Wien, Nordbergstraße 15,

A-1090 Vienna, Austria. WWW: http://www.mat.univie.ac.at/~kratt.

T. Rivoal, Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France.

WWW: http://www-fourier.ujf-grenoble.fr/~rivoal.