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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
\zetaq(s)=
\sum _{k=1} ^{\infty}qk
\sum _{d|k}ds-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,\zetaq(3),
\zetaq(5), ...,
\zetaq(M) is at least
c1M1/2,
where c1=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
c2log M, with
c2=0,5906.
For the same values of q, a similar lower bound for the values
\zeta1(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
E4(q) and
E6(q) is transcendental over Q for
any complex number q such that 0<|q|<1.
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