Automorphic Integrals with Rational Period Functions and Arithmetical Identities

TL;DR

This paper extends the equivalence between functional equations and arithmetical identities from modular cusp integrals to automorphic integrals with rational period functions on Hecke groups, broadening the scope of these classical results.

Contribution

It generalizes Heath's results to automorphic integrals on Hecke groups, establishing the equivalence between functional equations and arithmetical identities in this broader setting.

Findings

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

Extended classical results from modular groups to Hecke groups.

Provided new insights into the structure of automorphic integrals with rational period functions.

Abstract

In 1961, Chandrasekharan and Narasimhan showed that for a large class of Dirichlet series the functional equation and two types of arithmetical identities are equivalent. In 1992, Hawkins and Knopp proved a Hecke correspondence theorem for modular integrals with rational period function on theta group. Analogous to Chandrasekharan and Narasimhan, in 2015 Sister Ann M. Heath has shown that the functional equation in Hawkins and Knopp context and two type of arithmetical identities are equivalent. She considered the functional equation 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 extend the results of Sister Ann M. Heath to entire automorphic integrals involving rational period functions on discrete Hecke group.