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.

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., given a Calabi–Yau threefold M, they constructed another Calabi–Yau threefold, the “mirror” of M, whose properties can be used to enumerate the rational curves on M).

The purpose of the present article is to prove rather sharp integrality assertions for the Taylor coefficients of mirror maps coming from certain 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 and 2) encompass integrality results on these mirror maps which exist in the literature, improving and refining them in numerous instances.

In a sense, mirror maps can be viewed as higher order generalisations of certain classical modular forms (defined over various congruence sub-groups of SL_{2}(ℤ)), the latter appearing naturally at low order in Schwarz’s theory of hypergeometric functions (see [30]). For integers k ≥ 1 and N ≥ 1, let us define the power series

Date: November 8, 2009.

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, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

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

which converges for |z| < 1/N^{kN}. The function F (z) is solution of a hypergeometric differential equation of degree kN, which is a special case of (1.5) below. The equation has maximal unipotent monodromy (MUM) (in particular, the roots of the indicial equation at z = 0 are all 0). A basis of solutions with at most logarithmic singularities around z = 0 can then be obtained by Frobenius’ method; see [30]. In particular, there exists another solution of the form G(z) + log(z)F (z), where G(z) is holomorphic around 0,

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

with H_{n} := ∑_{i=1}^{n} \frac{1}{i} denoting the n-th harmonic number. In the context of mirror symmetry, the function q(z) := z exp(G(z)/F (z)) is usually called canonical coordinate, and its compositional inverse z(q) is the prototype of a mirror map.

In the case k = N = 2, one can express z(q) explicitly in terms of the Legendre function λ(q) := 16q ∏_{n=1}^{∞} ((1 + q^{2n})/(1 + q^{2n−1})), namely as z(q) = λ(q)/16, which is modular over Γ(2). Moreover, if k = 3 and N = 2, we have z(q) = λ(q)(1 − λ(q))/16. In particular, in both cases the power series q(z) and z(q) have integral Taylor coefficients. It is this fact that we will generalise in this paper. For other examples of mirror maps of modular origin, we refer to the discussion in [3, pp. 111–113] (and also for the importance of such facts in Diophantine approximation) and [20, Sec. 3].

The most famous (apparently non-modular) example of a mirror map arises in the case when N = 5 and k = 1, which was used in the epoch-making paper by the physicists Candelas et al. [7] in their study of 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 (see [23, 24, 28]).

The following conjecture belongs probably to the folklore of mirror symmetry theory.

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

Lian and Yau [20, Sec. 5, Theorem 5.5] proved this conjecture for k = 1 and any N which is a prime number. Zudilin [31, Theorem 3] extended their result by proving the conjecture for any k ≥ 1 and any N which is a prime power.

Our original goal was to settle Conjecture 1 in complete generality. In the present paper, we shall accomplish much more. We shall prove a more general conjecture by Zudilin [31] concerning the integrality of Taylor coefficients of a very large class of mirror maps (see Conjecture 2) and refinements of the corresponding integrality results in special cases. In the remainder of this introductory section, we describe these two sets of results (see Theorems 1 and 2). Their proofs will then be given in the subsequent sections.

1In the number-theoretic study undertaken in the present paper, we are interested in the integrality of the coefficients of (roots of) mirror maps z(q). In that context, the mirror map z(q) and the corresponding canonical coordinate q(z) 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 [19, Introduction].) We shall, in the sequel, formulate our integrality results exclusively for canonical coordinates, assuming tacitly that the reader keeps in mind that they automatically also hold for the corresponding mirror maps. It is also therefore that, by abuse of terminology, we shall often use the term “mirror map” for any canonical coordinate.

1.2. Zudilin’s conjecture. In order to state Zudilin’s conjecture, we need to introduce some notation.

For a positive integer N, let p_{1}, p_{2}, …, p_{ℓ} denote its distinct prime factors. We define the vectors of integers

(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}  (1.1)

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} ,  (1.2)

where α_{1} + α_{2} + ⋯ + α_{μ} = β_{1} + β_{2} + ⋯ + β_{η}. We put B_{1}(m) := 1 and H_{1}(m) := 0, and, if N ≥ 2,

B_{N}(m) := \frac{∏_{j=1}^{μ}(α_{j}m)!}{∏_{j=1}^{η}(β_{j}m)!}  (1.3)

and (^{2})

H_{N}(m) = ∑_{j=1}^{μ} α_{j}H_{α_{j}m} − ∑_{j=1}^{η} β_{j}H_{β_{j}m}.  (1.4)

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}},

and, correspondingly,

H_{4}(m) = 4H_{4m} − 2H_{2m} − 2H_{m}, H_{6}(m) = 6H_{6m} − 3H_{3m} − 2H_{2m} − H_{m}, H_{30}(m) = 30H_{30m} + 5H_{5m} + 3H_{3m} + 2H_{2m} − 15H_{15m} − 10H_{10m} − 6H_{6m} − 9H_{m}.

Given a vector N = (N_{1}, N_{2}, …, N_{k}) of positive integers, we construct the power series

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

and

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

2Zudilin’s definition [31, Eq. (5)] of the quantity H_{N}(m) (he writes D_{N}(m)) is different from (1.4).

We do not need it in our paper, but, for the sake of completeness, we prove the equivalence of the two definitions in Section 11.

It can be shown (see [31], taking into account Lemma 15 in Section 11) that the series F_{N}(z) and G_{N}(z) + log(z) F_{N}(z) are two solutions to the equation Ly = 0, where the hypergeometric differential operator L is defined by

L := (z \frac{d}{dz} )^{φ(N_{1})+···+φ(N_{k})} − C_{N}z ∏_{j=1}^{k} ∏_{i=1}^{φ(N_{j})} (z \frac{d}{dz} + \frac{r_{i,j}}{N_{j}} ) ,  (1.5)

where φ( .) is Euler’s totient function, C_{N} := ∏_{j=1}^{k} C_{N_{j}} with C_{N_{j}} := N_{j}^{φ(N_{j})}_{∏p|N_{j}}p^{φ(N_{j})/(p−1)}, and the r_{i,j} ∈ {1, 2, …, N_{j}} form the residue classes modulo N_{j} which are coprime to N_{j}. The differential equation Ly = 0 has MUM (^{3}) at the origin.

We can now state Zudilin’s conjecture from [31, p. 605]. (Zudilin’s formulation is different. That our formulation is equivalent with Zudilin’s follows from [31, Lemma 4], respectively (11.1), and from Lemma 15 in Section 11.)

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]].

It can be seen (see [31, paragraph above Theorem 2], respectively (1.6)–(1.8) below) that Conjecture 1 is the special case of the above conjecture where k is replaced by k · d(N) (with d(N) denoting the number of positive divisors of N), and where {N_{1}, …, N_{k·d(N)}} is the multiset (^{4}) in which each divisor of N appears exactly k times.

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}.

We claim that Conjecture 2 follows from the theorem below, which is the first main result of the paper. 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 1. 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 2, with details being filled in in Sections 4–8.

To see that Theorem 1 implies Conjecture 2, we observe that, by (1.4), ∑_{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})}.

Thus, z^{−1}q_{N}(z) is a product of series q_{L,N}(z), each one raised to an integer power. It follows that Conjecture 2 implies Theorem 1, as claimed.

3This 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 [8] and [13, Sec. 3].

4A multiset is a “set” where one allows repetitions of elements.

1.3. Stronger integrality properties in special cases. Let N_{1}, N_{2}, …, N_{k} be given positive integers (not necessarily distinct). In the setting of Section 1.2 with k replaced by k · (d(N_{1}) + d(N_{2}) + ⋯ + d(N_{k})), d(N) again denoting the number of positive divisors of N, we consider the special case where the vector N can be partitioned into k blocks, the j-th block consisting of all the positive divisors of N_{j}, j = 1, 2, …, k. It can be seen (cf. [31, paragraph above Theorem 2]) that the functions F_{N}(z), G_{N}(z) and G_{L,N}(z) then simplify to

∑_{m=0}^{∞} ( ∏_{j=1}^{k} \frac{(N_{j}m)!}{m!^{N_{j}}})z^{m},  (1.6)

∑_{m=1}^{∞} ( ∑_{j=1}^{k} N_{j}(H_{N_{j}m} − H_{m}))( ∏_{j=1}^{k} \frac{(N_{j}m)!}{m!^{N_{j}}})z^{m},  (1.7)

∑_{m=1}^{∞} H_{Lm}( ∏_{j=1}^{k} \frac{(N_{j}m)!}{m!^{N_{j}}})z^{m},  (1.8)

respectively. In order to simplify notation, for the remainder of this subsection we “redefine” N by letting N = (N_{1}, N_{2}, …, N_{k}), and we denote the series in (1.6), (1.7), and (1.8) respectively by F_{N}(z), G_{N}(z), and G_{L,N}(z). Accordingly, we define q_{N}(z) := z exp (G_{N}(z)/F_{N}(z)) and q_{L,N}(z) := exp (G_{L,N}(z)/F_{N}(z)).

For the mirror map q_{(N)}(z) arising for k = 1, physicists made the observation that, apparently,

(z^{−1}q_{(N)}(z))^{1/N} ∈ ℤ[[z]].  (1.9)

This was proved by Lian and Yau [22] for any prime number N, thus strengthening their result from [20] mentioned after Conjecture 1. The observation (1.9) leads naturally to the more general question of determining the largest integer V such that (z^{−1}q(z))^{1/V} ∈ ℤ[[z]] for the mirror map q(z) = q_{(N)}(z) in (1.9) or other mirror maps. (^{5}) While we are not able to give a precise answer, we shall prove the following result for a large class of mirror maps which, as we explain in [16], comes very close to being optimal for this class.

Theorem 2. For any integers N_{1}, …, N_{k} ≥ 1, let M_{N} = ∏_{i=1}^{k} N_{i}!. Furthermore, let Θ_{L} := L!/ gcd(L!, L! H_{L}) be the denominator of H_{L} when written as a reduced fraction.

Then, for all L ∈ \{1, 2, …, max(N_{1}, …, N_{k})\}, we have q_{L,N}(z)^{\frac{Θ_{L}}{M_{N}}} ∈ ℤ[[z]].

Remarks. (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 be improved when L = N_{1} = ⋯ = N_{k}. See [16] for such results.

5Let 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. [12, Lemma 5] for a simple proof based on Bézout’s lemma).

(b) It is natural to expect refinements of Theorem 1 in the spirit of Theorem 2. 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 14 in Section 5 that B_{N}(1) always divides B_{N}(m) for any m ≥ 1 and any N, where B_{N}(m) := ∏_{j=1}^{k} B_{N_{j}}(m). Our techniques enable us to deduce that q_{1,N}(z)^{1/B_{N}(1)} ∈ ℤ[[z]] (cf. Remark 1 in Section 2), which proves the above assertion that q_{1,(6)}(z)^{1/60} ∈ ℤ[[z]]. 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.

An outline of the proof of Theorem 2 is given in Section 3, with details being filled in in Sections 9–10.

Due to the equation

q_{(N,...,N)}(z) = zq_{N,(N,...,N)}(z)^{kN}q_{1,(N,...,N)}(z)^{−kN},  (1.10)

(with k occurrences of N in (N, …, N)), Theorem 2 has the following consequence for the mirror map q_{(N,...,N)}(z), thus improving significantly upon (1.9).

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

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

Remarks. (a) 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 (1.9) by a factor of 2.

(b) Also Corollary 1 can be improved. The corresponding result, which is optimal subject to a widely believed conjecture on harmonic numbers, is very technical. We refer the reader again to our article [16].

1.4. Method of proof. The basic idea is to transfer the original integrality assertions into a p-adic framework (by means of Lemma 8), to a point where we can employ Dwork’s theory of formal congruences. This theory seems to provide the most powerful tools available for attacking integrality assertions of the type discussed in this paper. We recall the corner stones of Dwork’s theory in Section 4.

We draw the reader’s attention to the fact that, while the general line of our approach follows that of previous authors (particularly [22]), there does arise a crucial difference (other than just technical complications): the reduction and rearrangement of the sums C(a + Kp) in Section 2, respectively C(a + Kp) in Section 3, via the congruence (2.3) require a new reduction step, namely Lemma 1, respectively Lemma 5. In fact, the proofs of these two lemmas form the most difficult parts of our paper. (In previous work, the use of (2.3) sufficed because the corresponding authors restricted themselves to N being prime or a prime power.)

1.5. Related work and perspectives. One of the goals of the present paper is to highlight number-theoretic phenomena in the context of mirror symmetry theory. Our analysis is clearly on the number theory side. Nevertheless, we hope that our results contribute to the clarification of such phenomena. On the other hand, it should not be hidden that integrality phenomena should also have intrinsic significance, within mirror symmetry. One such example is the (apparent) coincidence of coefficients in Yukawa couplings with numbers of certain rational curves (cf. [11, 21]; see also the discussion in [28, p. 49]).

In the context of our results, it is possible to prove similar, but strictly weaker, statements by means of methods of arithmetic geometry. This is the case for the preprint [29], which is an elaborate version of [14]. The mirror maps that are considered in that paper comprise ours. When both approaches apply simultaneously, our results in Theorems 1 and 2 are stronger than Theorem 2 in [29, Sec. 1.3]. Indeed, we prove that certain mirror maps have integral Taylor coefficients, while in [29] 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 [29]).

We also want to point out that the range of application of Dwork’s ideas is not restricted to p-adic functions in one variable. In [15], we extend Dwork’s theory as outlined in Section 4 to several variables. As applications, we obtain integrality properties of mirror maps in several variables arising in the context of the very general multivariable mirror maps coming from the Gelfand–Kapranov–Zelevinsky hypergeometric series (see [5, Sec. 7.1], [13] and [27, Sec. 8] for numerous examples related to Calabi–Yau manifolds which are complete intersections in products of weighted projective spaces). As a by-product, by appropriate specialisations, we are even able to prove (predicted) integrality of the Taylor coefficients of some mirror maps in one variable that do not fall under the scope of the results of the present paper, in particular many of those in the table presented in [1].

We therefore believe that Dwork’s methods provide valuable insight in integrality properties of mirror maps. Since the power and range of applicability of these methods have apparently not yet been exhausted, their development should be further pursued. In particular, we hope to be able to test them against the difficult number-theoretical problems raised by Yukawa couplings.

Finally, we point out that the quantity B_{N}(m) defined in (1.3), which is the crucial building block for the series F_{N}(z), G_{N}(z), and G_{L,N}(z), is in fact an integer for all N and m. This follows from the criterion [17] of Landau, which, applied to our case, says that, for a fixed N, B_{N}(m) is integral for all m if and only if

∑_{j=1}^{μ} ⌊α_{j}x⌋ − ∑_{j=1}^{η} ⌊β_{j}x⌋ ≥ 0

for all non-negative real numbers x. Indeed, the above quantity is exactly the quantity Δ(x) defined in Lemma 12 in Section 5, which plays a crucial role in the proof of Lemma 1 in Section 6, and, thus, in the proof of Theorem 1. More precisely, amongst its properties, we use many times the fact that it is weakly increasing on [0, 1). We point out that such “Landau functions” were introduced in the theory of mirror symmetry of Calabi– Yau threefolds by Rodriguez-Villegas [25], who used them to prove that there are exactly 14 hypergeometric functions whose coefficients can be written as integral quotients of factorials (after rescaling the variable z to Cz for some C) and with the MUM property at the origin. In [26], he also used Landau functions to characterise algebraic hypergeometric series whose coefficients can be written as integral quotients of factorials after rescaling: using the theory of Beukers and Heckman [6], he proved that such series are algebraic if and only if Δ(x) ∈ {0, 1}. Of course, it is not always possible to write the Taylor coefficients of a hypergeometric series as quotients of factorials.

It can be proved that amongst hypergeometric series whose Taylor coefficients are integral quotients of factorials, the weak increase of the associated Landau function Δ is equivalent to the MUM property. This is a consequence of more general results of Eric Delaygue, in a thesis currently being written under the supervision of the second author. Thus, from the point of view of mirror symmetry where the MUM property is essential, our Theorem 1 is best possible in that class of hypergeometric series. The question then naturally arises which properties the Landau function Δ must have such that the mirror map formally associated to it (by forming functions analogous to F_{N} and G_{N}) has integral Taylor coefficients (even when a mirror symmetry interpretation is not available; for example, when the MUM property does not hold). We believe that the following statement is true: the formal mirror map has integral Taylor coefficients if and only if Δ(x), defined on [0, 1), remains ≥ 1 after its first jump from 0. New ideas are necessary to solve this problem because the approach used in this paper does not work in this more general setting; indeed there are counterexamples to almost all our lemmas when Δ is not weakly increasing. This is currently under investigation by Eric Delaygue.

1.6. Structure of the paper. We now briefly review the organisation of the rest of the paper.

The proofs of our theorems being highly complex, we start with brief outlines of the proofs of Theorems 1 and 2 in Sections 2 and 3, respectively. These outlines reduce the proofs to a certain number of lemmas. The reduction is heavily based on Dwork’s theory of formal congruences, the relevant pieces of which being recalled in Section 4. The lemmas which are necessary for the proof of Theorem 1 are subsequently established in Sections 5–8, while those necessary for the proof of Theorem 2 are established in Sections 9–10.

2. Outline of the proof of Theorem 1

In this section, we provide a brief outline of the proof of Theorem 1. As we already said in the introduction, the proof follows the p-adic approach pioneered by Dwork [9, 10], of which we review its corner stones in Section 4. We show that this approach allows us to reduce the proof of Theorem 1 to Lemmas 1–3. These lemmas are subsequently proved in Sections 6–8, with four auxiliary lemmas being the subject of Section 5.

By Dwork’s Lemma given in Section 4 (or rather its consequence given in Lemma 10), 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}),  (2.1)

where B_{N}(m) := ∏_{j=1}^{k} B_{N_{j}}(m), the quantities B_{N_{j}}(m) being defined in (1.3). In view of Lemma 10, proving Theorem 1 is equivalent to proving that

C(a + Kp) ∈ pℤ_{p}  (2.2)

for all primes p and non-negative integers a and K with 0 ≤ a < p. 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}.  (2.3)

Applying this with J = La + Ljp, we get

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

This implies that

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}.  (2.4)

We now want to transform the sum on the right-hand side of (2.4) 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 6.

Lemma 1. 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}.  (2.5)

It follows from Eq. (2.4) and Lemma 1 that

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

(2.6)

We now use a combinatorial lemma due to Dwork (see [10, Lemma 4.2]) which provides an alternative way to write the sum on the right-hand side of (2.6): 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}, (2.7)

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} ∈ pℤ_{p}  (2.8)

for all m and s, because (2.6) and (2.7) would then imply that C(a+Kp) ∈ pℤ_{p}, as desired.

We will prove (2.8) in the following manner. The expression for S(a, K, s, p, m) is of the form considered in Proposition 1 in Section 4. The proposition will enable us to prove the following fact in Section 7.

Lemma 2. 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}.

Furthermore, in Section 8 we shall prove the following lemma.

Lemma 3. 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}.  (2.9)

It is clear that Lemmas 2 and 3 imply (2.8). This completes the outline of the proof of Theorem 1.

Remark 1. To prove the refinement announced at the end of the Introduction that q_{1,N}(z)^{1/B_{N}(1)} ∈ ℤ[[z]], by Lemma 10 we should show that C(a + Kp) ∈ pB_{N}(1)ℤ_{p} instead of the weaker (2.2), which means to show that

C(a + Kp) ≡ ∑_{j=0}^{K} B_{N}(a + jp)B_{N}(K − j)(H_{L(K−j)} − H_{⌊La/p⌋+Lj}) mod pB_{N}(1)ℤ_{p} (2.10)

instead of the weaker (2.4), that B_{N}(a + pj)(H_{j+⌊a/p⌋} − H_{j}) ∈ pB_{N}(1)ℤ_{p} instead of (2.5) (but this is trivial because H_{j+⌊a/p⌋} − H_{j} = 0), and that

B_{N}(m) (H_{Lmp^{s}} − H_{L⌊m/p⌋p^{s+1}}) ∈ \frac{B_{N}(1)}{p^{s}} ℤ_{p}  (2.11)

instead of the weaker (2.9). To establish (2.10) one would apply the same type of argument as the one establishing (3.3), however with Lemma 4 replaced by Lemma 14, the latter lemma being proved in Section 5. To prove (2.11), Lemma 14 must be used in (8.2).

3. Outline of the proof of Theorem 2

This section is devoted to an outline of the proof of Theorem 2, reducing it to Lemmas 5– 7. These lemmas are subsequently proved in Sections 9–10.

We follow the strategy that we used in Section 2 to prove Theorem 1; that is, by the consequence of Dwork’s Lemma given in Lemma 10, 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]].

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 10, 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.

Using this together with (2.3) specialised to J = La + Ljp, 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}. (3.3)

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 (3.3) is equal to 0, and therefore the congruence (3.3) holds trivially because C(0) = 0.

The analogue of Lemma 1 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 9.

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}.  (3.4)

We now do the same rearrangements as those after Lemma 1 to conclude that

C(a + Kp) ≡ − ∑_{s=0}^{r} ∑_{m=0}^{p^{r+1−s}−1} Y_{m,s} mod p\frac{M_{N}}{Θ_{L}} ℤ_{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} ∈ p\frac{M_{N}}{Θ_{L}} ℤ_{p},  (3.5)

then C(a + Kp) ∈ p\frac{M_{N}}{Θ_{L}} ℤ_{p}, as desired.

Now, the last assertion follows from the following two lemmas. Lemma 6 is the special case of Lemma 2 where the vector N is specialised in the way described at the beginning of Section 1.3 (in which case the quantity B_{N}(m) of Lemma 2 reduces to B_{N}(m), and hence S(a, K, s, p, m) to S(a, K, s, p, m)). On the other hand, Lemma 7 is the analogue of Lemma 3. Its proof can be found in Section 10.

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}.  (3.6)

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} .  (3.7)

It is clear that (3.6) and (3.7) imply (3.5). This completes the outline of the proof of Theorem 2.

4. Dwork’s theory of formal congruences

In this section, we review those aspects of Dwork’s theory on which the arguments of the proofs of Theorems 1 and 2 (see Sections 2 and 3) are based.

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

Lemma 8. 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 9 (“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 [18, 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 [20, 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 10. 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]].  (4.1)

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 (4.1) from the following crucial result, also due to Dwork (see [10, Theorem 1.1]).

Proposition 1 (“Dwork’s Formal Congruences Theorem”). Let A : ℤ_{≥0} → ℚ_{p}^{×}, g : ℤ_{≥0} → ℤ_{p} \ {0} be mappings such that

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

(ii) A(m) ∈ g(m)ℤ_{p};

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

\frac{A(v + up + np^{s+1})}{A(v + up)} − \frac{A(u + np^{s})}{A(u)} ∈ p^{s+1} \frac{g(n)}{g(v + up)} ℤ_{p}.  (4.2)

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

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

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

F (z^{p})F_{m,s+1}(z) − F (z)F_{m,s}(z^{p}) ∈ p^{s+1}g(m)ℤ_{p}[[z]],  (4.3)

or, equivalently,

∑_{j=mp^{s}}^{(m+1)p^{s}−1} (A(a + jp)A(K − j) − A(j)A(a + (K − j)p)) ∈ p^{s+1}g(m)ℤ_{p}  (4.4)

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

Remarks. (a) Dwork’s original theorem is in fact more general in that it contains families of functions A_{r} and g_{r}, r = 0, 1, 2, …, (which are all equal to A, respectively to g, in the above specialisation). Moreover, 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 (4.3) and (4.4), with O_{p} instead of ℤ_{p}. In our more restrictive setting, (4.3) and (4.4) hold because (p^{s+1}g(m)O_{p}) ∩ ℚ_{p} = p^{s+1}g(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(a + jp)A(K − j) − A(j)A(a + (K − j)p))  (4.5)

is exactly the (a + pK)-th Taylor coefficient of F (z^{p})F_{m,s+1}(z) − F (z)F_{m,s}(z^{p}), which explains the equivalence between the formal congruence (4.3) and the congruence (4.4). Note that in (4.5) the value of A at negative integers must be taken as 0.

(c) Most authors chose g(m) = 1 or a constant in m. We will use instead g(m) = A(m): this choice has already been made by Dwork in [9, Sec. 2, p. 37].

(d) Dwork also applied his methods to the problems considered in the present paper. Indeed, he proved a result, namely [10, 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 [31, Proposition 2] for details). This fact was used by the authors of [19, 20, 31] 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.

During the proofs of Lemma 2 in Section 7, 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 ℤ_{p}. In the following lemma, we collect the results on Γ_{p} that we shall need later on.

Lemma 11. (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 [31, Lemma 7] for (i) and [18, p. 71, Lemma 1.1] for (ii). □

5. Auxiliary lemmas

In this section, we establish three auxiliary results. The first one, Lemma 12, is required for the proof of Lemma 1 in Section 6, while the second one, Lemma 13, is required for the proof of Lemma 2 in Section 7. The third result, Lemma 14, justifies an assertion made in the Introduction (see item (b) in the remarks after Theorem 2). Moreover, the proofs of Lemmas 13 and 14 make themselves use of Lemma 12.

Lemma 12. 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 {2, 3, …, N}, we have Δ(r) ≥ 1.

Remark. 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}.  (5.1)

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 (5.1) (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 13. 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},  (5.2)

where B_{N}(m) is the quantity defined after (2.1).

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 (1.3) and Legendre’s formula v_{p}(n!) = ∑_{k=1}^{∞}⌊\frac{n}{p^{k}} ⌋), 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 (5.5)). 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, (5.3)

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 12(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, (5.4)

(The reader should note the difference to (5.3) 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 (5.4) 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}⌋ )).  (5.5)

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 12(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 (5.5) is non-negative, thus establishing (5.4) and also (5.3). This finishes the proof of the lemma. □

We conclude this section with a result which was announced in item (b) of the remarks after Theorem 2. It is used nowhere else, but we give it here 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 Remark 1 in Section 2).

Lemma 14. For any vector N and any integer m ≥ 1, we have that B_{N}(1) divides B_{N}(m), where B_{N}(m) is the quantity defined after (2.1).

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 12. 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 (partial) monotonicity of Δ described in Lemma 12(ii). □

6. Proof of Lemma 1

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.

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, without loss of generality, max(N_{1}, …, N_{k}) = N_{k}. Then, using again Lemma 12(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⌋ .

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 + v_{p}(Lj + ε)  (6.1)

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}}.

6.1. First step. We claim that

Δ_{k}(\frac{a + jp}{p^{ℓ}}) ≥ Δ_{k}(\frac{pj + pε/L}{p^{ℓ}}) .  (6.2)

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.  (6.3)

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 12(i),(ii) (with Δ = Δ_{k}) and (6.3) 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 (6.2) made at the beginning of this step.

6.2. 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} ) .  (6.4)

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. Furthermore, for ℓ ≤ d + 1, the denominator of \frac{βp^{d+1−ℓ}}{L} is obviously at most L. Since L ≤ N_{k}, it follows then from Lemma 12(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 (6.4) gives

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

This completes the proof of (6.1) and, hence, of Lemma 1.

7. Proof of Lemma 2

We want to use Proposition 1 with A(m) = g(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.

Condition (i) is true since B_{N}(0) = 1. Condition (ii) holds by the definitions of A(m) and g(m). To check that Condition (iii) holds is more involved. The proof will be decomposed in three steps. The reader should recall that

B_{N}(m) := ∏_{j=1}^{k} B_{N_{j}}(m),

where B_{N_{j}}(m) is given by (1.3), or, alternatively (cf. [31, Lemma 4], respectively (11.1) below) as

B_{N_{j}}(m) = C_{N_{j}}^{m} ∏_{ℓ=1}^{φ(N_{j})} \frac{(r_{ℓ,j}/N_{j})_{m}}{m!} ,  (7.1)

where C_{N_{j}} and the r_{ℓ,j}’s are defined as in Subsection 1.2. Expression (7.1) will be useful in the first step below, while the direct use of Expression (1.3) would lead to much more involved computations.

7.1. First step. Let us fix j ∈ {1, 2, …, k}. We set D_{N_{j}} := N_{j}^{−φ(N_{j})}C_{N_{j}}, 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}),  (7.2)

where O(R) denotes an element of Rℤ_{p}. To prove (7.2), we observe that (^{6})

\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,j} + (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})}} (7.3)

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

\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})).

6Identities (7.3) and (7.5) are immediate consequences of the alternative form (7.1) of B_{N_{j}}. Zudilin used them in his proof of the following stronger version of (7.2):

However, for this, he assumes that p divides N_{j} (see [31, Eq. (35)]). Here, we do not assume that p divides N_{j}, and therefore we obtain the weaker equality (7.2), which is fortunately enough for our purposes.

If v = 0, then (7.2) 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))^{φ(N_{j})} + O(p^{s+1})}

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

Hence,

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

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

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

which proves (7.2) because

\frac{1}{(v + up)(v − 1 + up) ⋯ (1 + up)} ∈ ℤ_{p}  (7.4)

and

\frac{D_{N_{j}}^{v} ∏_{ℓ=1}^{φ(N_{j})} ∏_{i=1}^{v} (r_{ℓ,j} + (i − 1)N_{j} + uN_{j}p)}{((v + up)(v − 1 + up) ⋯ (1 + up))^{φ(N_{j})}} = \frac{B_{N_{j}}(v + up)}{B_{N_{j}}(up)}.  (7.5)

A side result of (7.5) (which was actually used to prove (7.2)) is that

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

We deduce from this fact and from (7.2) 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}).  (7.6)

7.2. Second step. Let us fix j ∈ {1, 2, …, k}. 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_{p}(1 + α_{i,j}(up + np^{s+1}))}{∏_{i=1}^{η_{j}} Γ_{p}(1 + β_{i,j}(up + np^{s+1}))}  (7.7)

= (−1)^{μ_{j}−η_{j}} \frac{∏_{i=1}^{μ_{j}} Γ_{p}(1 + α_{i,j}up) + O(p^{s+1})}{∏_{i=1}^{η_{j}} Γ_{p}(1 + β_{i,j}up) + O(p^{s+1})}  (7.8)
= (−1)^{μ_{j}−η_{j}} \frac{∏_{i=1}^{μ_{j}} Γ_{p}(1 + α_{i,j}up)}{∏_{i=1}^{η_{j}} Γ_{p}(1 + β_{i,j}up)} (1 + O(p^{s+1}))  (7.9)
= \frac{B_{N_{j}}(up)}{B_{N_{j}}(u)} (1 + O(p^{s+1})),  (7.10)

where (i) of Lemma 11 is used to see (7.7) and (7.10), and (ii) is used for (7.8). Equation (7.9) holds because 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})).  (7.11)

7.3. Third step. We now multiply the right-hand and left-hand sides of (7.6) and (7.11). After simplification, we get

\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}),

(7.12)

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

If we compare (4.2) (with A(m) = g(m) = B_{N}(m)) and (7.12), 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}.  (7.13)

The first assertion in (7.13) can be rewritten as

\frac{B_{N}(u + np^{s})}{B_{N}(n)} · \frac{B_{N}(v + up)}{B_{N}(u)} ∈ ℤ_{p},  (7.14)

while the second assertion can be rewritten as

\frac{B_{N}(u + np^{s})}{B_{N}(n)} ∈ ℤ_{p}.  (7.15)

Now, the assertion (7.15) is the special case w = u, m = n and r = s of Lemma 13, while (7.14) follows from (7.15) combined with the special case w = v, m = u and r = 1 of Lemma 13.

This completes the proof of the lemma.

8. Proof of Lemma 3

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 (2.5) shows that we are in a very similar situation here. Indeed, we may derive (2.9) from Lemma 1. 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}.  (8.1)

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))} . (8.2)

Use of Lemma 1 then completes the proof.

9. Proof of Lemma 5

We follow the same approach as the one of the proof of Lemma 1 in Section 6. In particular, the first part below is completely parallel to the proof of Lemma 1. We nevertheless include it here for the sake of readability and for later reference. On the other hand, since Lemma 5 makes a stronger divisibility assertion than Lemma 1, much more work is needed to arrive there: the corresponding arguments form the contents of the second and third part of this proof.

We start again by observing that the assertion (3.4) 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}).

9.1. 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} .  (9.1)

(The reader should note the absence of the term M_{N}/Θ_{L} in comparison with (3.4).)

For the proof of (9.1), 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}} ⌋) .  (9.2)

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).  (9.3)

The lower bound on the right-hand side of (9.2) can, in fact, be simplified since 0 ≤ a < p; namely, we have

⌊\frac{a + pj}{p^{ℓ}} ⌋ = ⌊\frac{j}{p^{ℓ−1}} ⌋ .  (9.4)

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 (9.2), together with (9.4), 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}⌋) .

(9.5)

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}⌋ .  (9.6)

Clearly, we also have

⌊−\frac{ε}{Lp^{ℓ−1}} + \frac{β}{L}p^{d+1−ℓ}⌋ ≤ ⌊\frac{β}{L}p^{d+1−ℓ}⌋ .  (9.7)

If we use (9.6) and (9.7) in (9.5), 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}⌋) .  (9.8)

By the same argument as the one that we used in the second step of the proof of Lemma 1 in Section 6, the rational number \frac{βp^{d+1−ℓ}}{L} is not an integer. 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 (9.8) gives

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

This completes the proof of (9.3), and, hence, of (9.1).

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.  (9.9)

We now embark on the proof of (3.4) itself.

9.2. 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} ,  (9.10)

or, using (2.3) (in the other direction), equivalently

B_{N}(a)H_{La} ∈ \frac{M_{N}}{Θ_{L}} ℤ_{p} ,  (9.11)

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 (9.10), reduces to B_{N}(a)H_{⌊La/p⌋} ∈ pℤ_{p}, which is indeed true because of (9.1) with j = 0.

Now let p ≤ N_{k}. By Lemma 4 and the definition of Θ_{L}, our claim, this time in the form (9.11), holds for a = 1. So, let a ≥ 2 from now on.

In a similar way as we did for the expression in (9.1), we bound the p-adic valuation of the expression in (9.11) 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  (9.12)

≥ max {1, ⌊L/p⌋} + ∑_{i=1}^{k} v_{p}(N_{i}!) − ⌊log_{p} L⌋ − 1.  (9.13)

If p = 2, then we can continue the estimation (9.13) as

v_{2}(B_{N}(a)H_{La}) ≥ ∑_{i=1}^{k} v_{2}(N_{i}!) − ⌊log_{2} L⌋ = v_{2}(M_{N}/Θ_{L}),  (9.14)

where we used the simple fact v_{2}(H_{L}) = − ⌊log_{2} L⌋ 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

x ≥ ⌊log_{p} x⌋ + 2  (9.15)

for all integers x ≥ 2 and primes p ≥ 3. Thus, in the case that L ≥ 2p, the estimation (9.13) 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 (9.11) in this case. If p ≤ L < 2p, then the estimation (9.13) 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 (9.11) in this case also. Finally, if L < p, it follows from (9.13) that

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

implying (9.11) also in this final case. Thus, (9.10) is established.

9.3. Third part: the case j > 0. Now let j > 0. If p > N_{k} = max(N_{1}, …, N_{k}), then (3.4) reduces to

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

which is again true because of (9.1).

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 (9.1), we bound the p-adic valuation of the expression in (3.4) 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})  (9.16)

≥ ⌊\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}} ⌋)  (9.17)

≥ 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)⌋}} ⌋  (9.18)

≥ 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)⌋}} ⌋  (9.19)

≥ max \{1, ⌊L/p⌋\} + ⌊log_{p} j⌋ − ⌊log_{p} (Lj + ⌊La/p⌋)⌋ + ∑_{i=1}^{k} ∑_{ℓ=1}^{∞} ⌊\frac{N_{i}}{p^{ℓ}} ⌋  (9.20)

≥ 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}!)  (9.21)
≥ max {1, ⌊L/p⌋} − ⌊log_{p} L⌋ − 1 + v_{p}(M_{N}).  (9.22)

Here, we used (9.9) in order to get (9.16). To get (9.18), we used the inequalities

⌊\frac{N_{k}a}{p}⌋ ≥ ⌊\frac{N_{k}}{p} ⌋ ≥ max \{1, ⌊L/p⌋\}  (9.23)

and

T_{1} + v_{p}(H_{Lj+⌊La/p⌋} − H_{Lj}) ≥ 0.  (9.24)

To get (9.19), 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.  (9.25)

Next, to get (9.20), we used

⌊\frac{N_{i}}{p^{ℓ}} · \frac{a + pj}{p^{⌊log_{p}(a+pj)⌋}} ⌋ ≥ ⌊\frac{N_{i}}{p^{ℓ}} ⌋ .  (9.26)

To get (9.21), we used

⌊log_{p} (Lj + ⌊La/p⌋)⌋ ≤ ⌊log_{p} L⌋ + ⌊log_{p} (j + \frac{1}{L}⌊La/p⌋)⌋ + 1.  (9.27)

Finally, we used \frac{1}{L}⌊La/p⌋ < 1 in order to get (9.22).

If we now repeat the arguments after (9.13), then we see that the estimation (9.22) implies

v_{p}(B_{N}(a + pj) (H_{Lj+⌊La/p⌋} − H_{Lj}) ) ≥ v_{p}(M_{N}/Θ_{L}).  (9.28)

This almost proves (3.4), our lower bound on the p-adic valuation of the number in (3.4) is just by 1 too low.

In order to establish that (3.4) is indeed correct, we assume by contradiction that all the inequalities in the estimations leading to (9.22) and finally to (9.28) are in fact equalities. In particular, the estimations in (9.23) 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 (9.13) that led us from (9.22) to (9.28), 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})  (9.29)

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 (9.15) with x = ⌊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 (9.29) is satisfied, as desired. We now show that (9.29) holds as well in Cases 1–4, thus completing the proof of (3.4).

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 (9.25).

On the other hand, if p = 2 and L = 3 then, because of equality in the second estimation in (9.23), 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 (9.25), 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 (9.26) 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 (9.25), 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)⌋}} ⌋ .  (9.30)

If f = 1, then we obtain from (9.30) 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 (9.26) with ℓ = 1.

On the other hand, if f ≥ 2, then we obtain from (9.30) 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 (9.26) 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 (9.25), 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.

10. Proof of Lemma 7

We proceed in the same way as in the proof of Lemma 3 in Section 8. Again, the claim is trivially true if p divides m, so that we may 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 (3.4) shows that we are in a very similar situation here. Indeed, we may derive (3.7) from Lemma 5. In order to see this, we use (8.1) to deduce

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.

11. The equivalence of Zudilin’s and our definition of H_{N}(m)

Zudilin’s definition of the quantity H_{N}(m) deviates from (1.4). In this final section, we show that our definition is equivalent to Zudilin’s.

Lemma 15. 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 (1.1) and (1.2), respectively). Then

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

where H(x, m) := ∑_{n=0}^{m−1} \frac{1}{x+n}, and where r_{j} ∈ \{1, 2, …, N\} form the residue classes modulo N which are coprime to N. As before, φ( .) denotes Euler’s totient function.

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)},  (11.1)

where Γ(x) denotes the gamma function. This generalises Zudilin’s identity (1.3) to real values of m. We essentially extend his proof to real m, using the well-known formula [4, 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),  (11.2)

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 (11.1) 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 (11.2) 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)}, (11.3)

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),  (11.4)

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 (11.4), we see that the exponent of p_{j} in the expression (11.3) 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 (11.3), we arrive at the right-hand side of (11.1).

Now, let us call b(m) the function defined by both sides of (11.1), and let ψ(x) = Γ′(x)/Γ(x) be the digamma function. We will use the well-known property (see [4, 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 (11.1), 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}

= H_{N}(m),  (11.5)

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 (11.1), 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).  (11.6)

The lemma follows by equating the expressions (11.5) and (11.6) obtained for b′(m)/b(m).

□

Acknowledgements

The authors are extremely grateful to Alessio Corti and Catriona Maclean for illuminating discussions concerning the geometric side of our work. They are furthermore indebted to the anonymous referees for their criticism and suggestions, which helped to improve the presentation of the material considerably.

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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.