On the binary additive divisor problem in mean

TL;DR

This paper investigates the mean value of the classical additive divisor problem, providing bounds related to Motohashi's main term and extending results to shifted convolution sums of cusp form coefficients.

Contribution

It offers new upper bounds for the mean value of the additive divisor problem and extends these bounds to shifted convolution sums of Fourier coefficients.

Findings

Upper bound for the mean value with Motohashi's main term

Upper bound for the case with Atkinson's main term

Extension of bounds to shifted convolution sums of cusp form coefficients

Abstract

We study a mean value of the classical additive divisor problem. The main term we are interested in here is the one by Motohashi, but we also give an upper bound for the case where the main term is that of Atkinson. Furthermore, we point out that the proof yields an analogous upper bound for a shifted convolution sum over Fourier coefficients of a fixed holomorphic cusp form in mean.