Arithmetic properties of the Herglotz function
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 1cocycles for the modular group $\mathrm{PSL}(2,\mathbb{Z})$.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.15805
 Bibcode:
 2020arXiv201215805R
 Keywords:

 Mathematics  Number Theory
 EPrint:
 18 pages