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

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) σ{-r_1}(n_1) σ{-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 $σ_{a}(x)$ 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}.