Convolution identities for complex-indexed divisor functions and modular graph functions

TL;DR

This paper derives exact identities for complex-indexed divisor sums expressed through hypergeometric functions, connecting them to Fourier coefficients of Hecke cusp forms and modular graph functions, expanding previous divisor function identities.

Contribution

It introduces new identities for complex-indexed divisor sums involving hypergeometric functions, linking them to modular forms and L-values, extending prior work on integer indices.

Findings

Expressed divisor sum identities in terms of Fourier coefficients of Hecke cusp forms.

Connected divisor sums to L-values and modular graph functions.

Used trace formulae and Estermann zeta functions in proofs.

Abstract

We find exact identities for sums of the form \begin{equation*}\label{eq:convsumabs} \sum_{\stackrel{n_1+n_2 = n}{n_1 \in \mathbb{Z} \setminus \{ 0, n \} }} Q(n_1,n_2) \sigma_{-r_1}(n_1) \sigma_{-r_2}(n_2), \end{equation*} where $n\in\mathbb{N}$, $r_1,r_2\in\mathbb{C}$, $Q$ is a combination of hypergeometric functions, and $\sigma_{a}(x)$ denotes the divisor function. Specifically, we find that they can be expressed in terms of Fourier coefficients of Hecke cusp forms weighted by their $L$-values. This result expands upon previous work with Radchenko in which such identities were found for divisor functions with even integer index \cite{FKLR} and encompasses results of Jacobi \cite{motohashi1994binary} and Diamantis and O'Sullivan in \cite{diamantis2010kernels, o2023identities} for divisor functions with odd integer index. The proof of our result expresses these sums in terms of Estermann zeta functions and uses trace formulae. In addition, we use a regularization of divergent convolution sums to provide a mathematical explanation for $L$-values (non-critical in the sense of Deligne) appearing in modular graph functions \cite{DKS2021_2}.