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

Ksenia Fedosova, Kim Klinger-Logan

TL;DR

This work extends convolution identities for divisor sums to complex indices by linking shifted divisor convolutions to Fourier coefficients of Hecke cusp forms weighted by non-critical $L$-values. By expressing the sums through Estermann zeta functions and deploying Petersson/Motohashi-type trace techniques, the authors derive a main identity that decomposes into cusp-form $L$-value terms plus carefully regularized boundary contributions. A novel hypergeometric regularization framework justifies the appearance of $L$-values in modular graph function contexts and handles divergent sums that naturally arise there. The results subsume and generalize prior identities (including FKLR and Jacobi–Diamantis–O’Sullivan lineages), provide explicit closed-form expressions in many parameter regimes, and illuminate when $L$-values appear in non-critical settings with modular-graph-function relevance.

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}.

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

TL;DR

This work extends convolution identities for divisor sums to complex indices by linking shifted divisor convolutions to Fourier coefficients of Hecke cusp forms weighted by non-critical -values. By expressing the sums through Estermann zeta functions and deploying Petersson/Motohashi-type trace techniques, the authors derive a main identity that decomposes into cusp-form -value terms plus carefully regularized boundary contributions. A novel hypergeometric regularization framework justifies the appearance of -values in modular graph function contexts and handles divergent sums that naturally arise there. The results subsume and generalize prior identities (including FKLR and Jacobi–Diamantis–O’Sullivan lineages), provide explicit closed-form expressions in many parameter regimes, and illuminate when -values appear in non-critical settings with modular-graph-function relevance.

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 , , is a combination of hypergeometric functions, and denotes the divisor function. Specifically, we find that they can be expressed in terms of Fourier coefficients of Hecke cusp forms weighted by their -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 -values (non-critical in the sense of Deligne) appearing in modular graph functions \cite{DKS2021_2}.
Paper Structure (32 sections, 8 theorems, 165 equations)

This paper contains 32 sections, 8 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. $d$ is a positive integer
  4. $d + r_1$ is an integer
  5. $d$ is a positive half-integer and $r_1$ and $r_2$ are integers of opposite parity
  6. Divergent series and modular graph functions
  7. Preliminaries
  8. Divisor function
  9. Estermann zeta function
  10. $L$-functions
  11. Bessel functions of the first type
  12. Hypergeometric functions
  13. Proofs
  14. Preliminary Results
  15. Definition of $\mathcal{E}(s)$ and its poles in the strip $\operatorname{Re}(s) < \operatorname{Re} (d)+1$
  16. ...and 17 more sections

Key Result

Theorem A

Let $n \in \mathbb{N}$, let $k$ be an even integer with $k \ge 6$ and let $r_1, r_2, d \in \mathbb{C}$ be such that Then, as an absolutely convergent series, where the summation runs over $\mathcal{F}_k$ a space of normalized Hecke cusp forms $f(z)=\sum_{n\geq 1}a_f(n)e^{2\pi i n z}$ of weight $k$, $L(\cdot, f)$ denotes the $L$-function associated to $f$, and $\langle \cdot, \cdot \rangle$ is t

Theorems & Definitions (15)

  • Theorem A
  • Theorem B
  • Lemma 2.1: Petersson trace formula
  • Proposition 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • proof
  • Lemma 3.3
  • ...and 5 more