An extension of Rohrlich's Theorem to the $j$-function

TL;DR

This paper extends Rohrlich's Theorem to compute inner products involving the $j$-function and meromorphic functions, revealing that their generating functions form part of a weight two modular object.

Contribution

It introduces an extension of Rohrlich's Theorem to include the $j$-function and connects the resulting generating functions to modular objects.

Findings

Extended Rohrlich's Theorem to the $j$-function

Established the modular nature of generating functions for inner products

Provided new tools for analyzing meromorphic functions and modular forms

Abstract

In this paper, we extend Rohrlich's Theorem on the integral of logarithms of meromorphic functions to compute the inner product between such functions and polynomials in the $j$-function. We then show that the generating function for these inner products with $j_n$ is part of a weight two modular object.