On the irrationality measure for a \(q\)-analogue of \(\zeta(2)\).

*(English. Russian original)*Zbl 1044.11067
Sb. Math. 193, No. 8, 1151-1172 (2002); translation from Mat. Sb. 193, No. 8, 49-70 (2002).

Given some \(p\in\mathbb{Z}\setminus\{0, \pm 1\}\), let \(q=1/p\). We may consider the series
\[
\zeta_q(2)= \sum_{n=1}^{\infty} \frac{q^n}{(1-q^{n})^2} =\sum_{n=1}^{\infty} \frac{p^n}{(p^{n}-1)^2}.
\]
The author proves the best currently known upper bound for the irrationality measure \(\mu\) of \(\zeta_q(2)\): for all \(p\in\mathbb{Z}\setminus\{0, \pm 1\}\), we have
\[
\mu(\zeta_q(2))\leq 4.07869375...
\]
In particular, \(\zeta_q(2)\) is not a Liouville number, which is an important improvement of the much weaker result deduced from Nesterenko’s theorem on the algebraic independence of values of modular forms.

To prove his result, the author first introduces a basic hypergeometric series with a particular form which enables him to construct rational linear forms in 1 and \(\zeta_q(2)\). He then adapts the (by now well-known) methods used to deal with ordinary hypergeometric series. He also adapts Rhin & Viola’s group method to eliminate big common factors from the coefficients of the rational linear forms.

Finally, let us emphasize that the notation \(\zeta_q(2)\) is motivated by the fact that \[ \lim_{q\to 1} (1-q)^2 \zeta_q(2)= \sum_{n=1}^{\infty} \frac{1}{n^2}=\zeta(2) \] and the author also shows that his approximations to \(\zeta_q(2)\) “tend” to Apéry’s well-known approximations to \(\zeta(2)\).

To prove his result, the author first introduces a basic hypergeometric series with a particular form which enables him to construct rational linear forms in 1 and \(\zeta_q(2)\). He then adapts the (by now well-known) methods used to deal with ordinary hypergeometric series. He also adapts Rhin & Viola’s group method to eliminate big common factors from the coefficients of the rational linear forms.

Finally, let us emphasize that the notation \(\zeta_q(2)\) is motivated by the fact that \[ \lim_{q\to 1} (1-q)^2 \zeta_q(2)= \sum_{n=1}^{\infty} \frac{1}{n^2}=\zeta(2) \] and the author also shows that his approximations to \(\zeta_q(2)\) “tend” to Apéry’s well-known approximations to \(\zeta(2)\).

Reviewer: Tanguy Rivoal (Caen)