A Dirichlet character analogue of Ramanujan's formula for odd zeta values

TL;DR

This paper generalizes Ramanujan's formula for odd zeta values using Dirichlet characters, extending previous series identities and deriving new relations for L-functions, including a novel identity for L(1/3, χ).

Contribution

It introduces a Dirichlet character analogue of a Ramanujan-type identity for odd zeta values for all natural N and integer h, expanding the scope of previous series transformations.

Findings

Derived a Dirichlet character analogue of Ramanujan's identity.

Established a two-variable generalization of Ramanujan's identity for ζ(2m+1).

Proved a new identity for L(1/3, χ) similar to Ramanujan's for ζ(1/2).

Abstract

In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, $$ \sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1}, $$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$. Recently, Dixit and the last author pointed out that this series has already been present in the Lost Notebook of Ramanujan with a more general form. Although, Ramanujan did not provide any transformation identity for it. In the same paper, Dixit and the last author found an elegant generalization of Ramanujan's celebrated identity for $\zeta(2m+1)$ while extending the results of Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored another extended version of the aforementioned series, namely, $$\sum_{r=1}^{q}\sum_{n=1}^{\infty} \frac{\chi(r)n^{N-2h}{\exp\left(-\frac{r}{q}n^N x\right)}}{1-\exp({-n^N x})},$$ where $\chi$ denotes a Dirichlet character modulo $q$, $N\in 2\mathbb{N}$ and with some restriction on the variable $h$. In the current paper, we investigate the above series for {\it any} $N \in \mathbb{N}$ and $h \in \mathbb{Z}$. We obtain a Dirichlet character analogue of Dixit and the last author's identity and there by derive a two variable generalization of Ramanujan's identity for $\zeta(2m+1)$. Moreover, we establish a new identity for $L(1/3, \chi)$ analogous to Ramanujan's famous identity for $\zeta(1/2)$.