A new Ramanujan-type identity for $L(2k+1,\chi_1)$

TL;DR

This paper extends Ramanujan-type identities from odd zeta values to $L$-series with principal characters, introduces new Ramanujan-type polynomials, and establishes related identities and analogues inspired by classical and recent work.

Contribution

It derives a new Ramanujan-type identity for $L(2k+1, \, \chi_1)$, extending previous identities for zeta values and introducing novel Ramanujan-type polynomials and identities.

Findings

Derived a new identity for $L(2k+1, \chi_1)$

Discovered a new family of Ramanujan-type polynomials

Established a character analogue of Grosswald's identity

Abstract

One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan's idea, we rediscovered a Ramanujan-type identity for $\zeta(2k+1)$ that was first established by Malurkar and later by Berndt using different techniques. In the current paper, we extend the aforementioned identity of Malurkar and Berndt to derive a new Ramanujan-type identity for $L(2k+1, \chi_1)$, where $\chi_1$ is the principal character modulo prime $p$. In the process, we encounter a new family of Ramanujan-type polynomials and we notice that a particular case of these polynomials has been studied by Lal\'{i}n and Rogers in 2013. Furthermore, we establish a character analogue of Grosswald's identity and a few more interesting results inspired from the work of Gun, Murty and Rath.