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Inst. Math. Jussieu **5** (2006), 53-79,
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# Séries hypergéométriques basiques,
*q*-analogues des valeurs de la fonction zêta et
séries d'Eisenstein

### (26 pages)

**English Abstract.**
We study the arithmetic properties of certain *q*-analogues
of values *\zeta*(*j*) of
the Riemann zeta function, in particular of the values of the functions
*\zeta*_{q}(*s*)=
*\sum* _{*k*=1} ^{*\infty*}*q*^{k}
*\sum* _{*d*|*k*}*d*^{s-1},
*s*=1,2,...,
where *q* is a complex number
with |*q*|<1. The main theorem of this article is that,
if *1/q* is an integer different from
-1 and 1, and if *M*
is a sufficiently large odd integer, then the dimension of the vector
space over **Q** which is spanned by 1,*\zeta*_{q}(3),
*\zeta*_{q}(5), ...,
*\zeta*_{q}(*M*) is at least
*c*_{1}*M*^{1/2},
where *c*_{1}=0,3358.
This result can be regarded as a *q*-analogue of
the result [14,2] that the dimension of the vector
space over **Q** which is spanned by 1,*\zeta*(3),
*\zeta*(5), ..., *\zeta*(*M*) is at least
*c*_{2}log *M*, with
*c*_{2}=0,5906.
For the same values of *q*, a similar lower bound for the values
*\zeta*_{1}(*s*)
at even integers *s* provides a new proof of
a special case of a result of Bertrand [*Bull. Soc. Math. France*
**104** (1976), 309-321] saying that one of the two Eisenstein series
*E*_{4}(*q*) and
*E*_{6}(*q*) is transcendental over **Q** for
any complex number *q* such that 0<|*q*|<1.

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