Shifted convolution sums motivated by string theory

TL;DR

This paper rigorously evaluates shifted convolution sums of divisor functions motivated by string theory, deriving exact identities and conjecturing the vanishing of certain sums related to Fourier coefficients in string scattering amplitudes.

Contribution

It generalizes previous formal arguments to rigorously evaluate specific shifted convolution sums of divisor functions and proposes new conjectures on their vanishing.

Findings

Derived exact identities for shifted convolution sums of divisor functions.

Established rigorous evaluation methods for sums related to string theory amplitudes.

Conjectured the vanishing of similar shifted sums beyond those previously studied.

Abstract

In \cite{CGPWW2021}, it was conjectured that a particular shifted sum of even divisor sums vanishes, and in \cite{SDK}, a formal argument was given for this vanishing. Shifted convolution sums of this form appear when computing the Fourier expansion of coefficients for the low energy scattering amplitudes in type IIB string theory \cite{GMV2015} and have applications to subconvexity bounds of $L$-functions. In this article, we generalize the argument from~\cite{SDK} and rigorously evaluate shifted convolution of the divisor functions of the form $\displaystyle \sum_{\stackrel{n_1+n_2=n}{n_1, n_2 \in \mathbb{Z} \setminus \{0\}}} \sigma_{k}(n_1) \sigma_{\ell}(n_2) |n_1|^R $ and $\displaystyle \sum_{\stackrel{n_1+n_2=n}{n_1, n_2 \in \mathbb{Z} \setminus \{0\} }} \sigma_{k}(n_1) \sigma_{\ell}(n_2) |n_1|^Q\log|n_1| $ where $\sigma_\nu(n) = \sum_{d \divides n} d^\nu$. In doing so, we derive exact identities for these sums and conjecture that particular sums similar to but different from the one found in \cite{CGPWW2021} will also vanish.