Extended Higher Herglotz function \textup{II}

TL;DR

This paper extends the theory of Herglotz functions, connecting it to Ramanujan's formulas and introducing a character analogue with a new functional equation, thus broadening the mathematical framework and applications.

Contribution

It generalizes Herglotz function theory, links it to Ramanujan's identities, and introduces a novel character analogue with a functional equation.

Findings

Derived an identity related to Ramanujan's formula for ζ(2m+1)

Connected Ramanujan's formula to Herglotz function theory

Introduced and initiated the theory of a character analogue of Herglotz functions

Abstract

Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for $\zeta(2m+1)$. We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.