A unified approach to Rohrlich-type divisor sums

TL;DR

This paper introduces a unified method for analyzing Rohrlich-type divisor sums across various congruence subgroups, unifying multiple results and providing new applications in modular form theory.

Contribution

It develops a systematic approach that unifies existing results on Rohrlich-type divisor sums and extends their applications to new formulas and proofs in the theory of modular forms.

Findings

Unified framework for Rohrlich-type divisor sums

Generalized Rohrlich's formula to arbitrary levels

Alternative proof of twisted trace decomposition

Abstract

We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups $\Gamma_0(N)$. Our main theorem unifies various results from the literature, and its significance is illustrated through the following five applications: (1) the valence formula, (2) a natural generalization of classical Rohrlich's formula to level $N$, (3) an explicit version of the theorem by Bringmann-Kane-L\"{o}brich-Ono-Rolen, (4) an extension of the generalized Rohrlich formula proposed by Bringmann-Kane, and (5) an alternative proof of the decomposition formula for twisted traces of CM values of weight 0 Eisenstein series.