Koshliakov zeta functions I: Modular Relations

TL;DR

This paper explores Koshliakov's generalizations of the Riemann zeta function, deriving new modular relations that extend Ramanujan's formulas and provide novel identities and representations for specific zeta values.

Contribution

It introduces two new modular relations for Koshliakov zeta functions, extending classical results and generating infinite families of identities for each positive real parameter p.

Findings

Derived a generalization of Ramanujan's formula for ζ(2m+1)

Extended a modular relation from Ramanujan's Lost Notebook

Obtained a new representation for ζ(4m+3)

Abstract

We examine an unstudied manuscript of N.~S.~Koshliakov over $150$ pages long and containing the theory of two interesting generalizations $\zeta_p(s)$ and $\eta_p(s)$ of the Riemann zeta function $\zeta(s)$, which we call \emph{Koshliakov zeta functions}. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each $p\in\mathbb{R^{+}}$. The first one is a generalization of Ramanujan's famous formula for $\zeta(2m+1)$ and the second is an elegant extension of a modular relation on page $220$ of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for $\zeta(4m+3)$.