English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

It is pointed out that the generalized Lambert series$\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied byKanemitsu, Tanigawa and Yoshimoto can be found on page $332$ of Ramanujan'sLost Notebook in a slightly more general form. We extend an importanttransformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto byremoving restrictions on the parameters $N$ and $h$ that they impose. From ourextension we deduce a beautiful new generalization of Ramanujan's famousformula for odd zeta values which, for $N$ odd and $m>0$, gives a relationbetween $\zeta(2m+1)$ and $\zeta(2Nm+1)$. A result complementary to theaforementioned generalization is obtained for any even $N$ and$m\in\mathbb{Z}$. It generalizes a transformation of Wigert and can be regardedas a formula for $\zeta\left(2m+1-\frac{1}{N}\right)$. Applications of thesetransformations include a generalization of the transformation for thelogarithm of Dedekind eta-function $\eta(z)$, Zudilin- and Rivoal-type resultson transcendence of certain values, and a transcendence criterion for Euler'sconstant $\gamma$.

Generalized Lambert series and arithmetic nature of odd zeta values