Generalized Lambert series and arithmetic nature of odd zeta values

TL;DR

This paper extends Lambert series transformations, leading to new relations between odd zeta values and generalizations of Ramanujan's formulas, with implications for transcendence and special constants.

Contribution

It generalizes Lambert series transformations by removing parameter restrictions, resulting in new formulas linking odd zeta values and applications in transcendence theory.

Findings

New relation between (2m+1) and (2Nm+1) for odd N

Generalization of Ramanujan's formula for odd zeta values

Transcendence criteria for Euler's constant mma

Abstract

It is pointed out that the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page $332$ of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters $N$ and $h$ that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for $N$ odd and $m>0$, gives a relation between $\zeta(2m+1)$ and $\zeta(2Nm+1)$. A result complementary to the aforementioned generalization is obtained for any even $N$ and $m\in\mathbb{Z}$. It generalizes a transformation of Wigert and can be regarded as a formula for $\zeta\left(2m+1-\frac{1}{N}\right)$. Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function $\eta(z)$, Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant $\gamma$.