Arithmetic properties of the Herglotz function

TL;DR

This paper explores the properties of functions introduced by Herglotz, revealing their special values, functional equations, and connections to deep number theory conjectures and modular group cohomology.

Contribution

It provides new insights into the arithmetic and functional properties of Herglotz functions, linking them to Stark's conjecture and modular cocycles.

Findings

Special values expressed as dilogarithms and logarithms

Functional equations from Hecke operators

Connections to Stark's conjecture and modular cocycles

Abstract

In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark's conjecture. We also discuss connections with 1-cocycles for the modular group $\mathrm{PSL}(2,\mathbb{Z})$.