Automorphic Integrals with Log-polynomial Period Functions and Arithmetical Identities

TL;DR

This paper extends the connection between automorphic integrals, arithmetical identities, and functional equations from the modular group to Hecke groups, focusing on log-polynomial period functions.

Contribution

It generalizes previous results to automorphic integrals on Hecke groups with log-polynomial period functions, establishing new equivalences with arithmetical identities.

Findings

Established equivalence between arithmetical identities and functional equations for automorphic integrals on Hecke groups.

Extended Heath's results from modular cusp integrals to automorphic integrals with rational period functions.

Connected arithmetical identities to automorphic integrals involving log-polynomial period functions.

Abstract

Building on the works of S. Bochner on equivalence of modular relation with functional equation associated to the Dirichlet series, K. Chandrasekharan and R. Narasimhan obtained new equivalences between the functional equation and some arithmetical identities. Sister Ann M. Heath considered the functional equation in the Hawkins and Knopp context and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we use techniques of Chandrasekharan and Narasimhan and extend the results of Sister Ann M. Heath to entire automorphic integrals involving rational period functions on discrete Hecke group. Moreover, we establish equivalence of two arithmetical identities with a functional equation associated with automorphic integrals involving log-polynomial-period functions on the Hecke groups.