Generalized Lambert series and arithmetic nature of odd zeta values

It is pointed out that the generalized Lambert series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{n^{N-2h}}{e^{n^{N}x}-1}}$ 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 $\zeta (2m+1-\frac{1}{N})$. 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 $\gamma$.