Extended higher Herglotz functions I. Functional equations

TL;DR

This paper introduces the extended higher Herglotz function, explores its functional equations, and connects it to generalized Lambert series and polylogarithm integrals, advancing understanding in algebraic and analytic number theory.

Contribution

It defines the extended higher Herglotz function, derives its functional equations, and links it to generalized Lambert series and polylogarithm integrals, extending previous work by Zagier and Vlasenko.

Findings

Derived two types of functional equations for the extended higher Herglotz function.

Established relations between the function and generalized Lambert series.

Obtained asymptotic expansions of the extended higher Herglotz function.

Abstract

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{Herglotz function}. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F(x)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $\mathscr{F}_{k,N}(x)$, an extension of the Herglotz function which also subsumes \emph{higher Herglotz function} of Vlasenko and Zagier. We call it the \emph{extended higher Herglotz function}. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by $\mathscr{F}_{k,N}(x)$. Radchenko and Zagier gave a beautiful relation between the integral $\displaystyle\int_{0}^{1}\frac{\log(1+t^x)}{1+t}\, dt$ and $F(x)$ and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between $\mathscr{F}_{k,N}(x)$ and a generalization of the above integral involving polylogarithm. The asymptotic expansions of $\mathscr{F}_{k, N}(x)$ and some generalized Lambert series are also obtained along with other supplementary results.