Generalized Lambert series and arithmetic nature of odd zeta values

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 $ζ(2m+1)$ and $ζ(2Nm+1)$. 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 $ζ\left(2m+1-\frac{1}{N}\right)$. Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function $η(z)$, Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant $γ$.