Modular Integrals and Their Mellin Transforms

Abstract

My purpose is to give a succinct and readable account of recent developments in the theory of modular integrals (with associated rational period functions) and the Mellin transforms of these. As I have already given one such exposition [13] which emphasizes the rational period functions on the modular group T(l) at the expense of the modular integrals (and their Mellin transforms), the present article will redress the balance, dealing mainly with the latter and putting aside discussion of rational period functions per se, whenever possible. Unavoidably there is a good bit of overlap between the present note and [13], to which it should be regarded as supplementary. Proofs are - or are to be - given elsewhere [4,10,11,12]. All of the results presented here can be generalized to accommodate (reasonably) general multiplier systems, but for the sake of clarity we restrict attention to multiplier system identically one.