On the general additive divisor problem

TL;DR

This paper establishes a new upper bound for the sum of error terms in the additive divisor problem for general k, improving previous bounds for the case k=3 when H is at least the square root of N.

Contribution

It provides a novel upper bound for the sum of error terms in the additive divisor problem for all k ≥ 3, extending and improving previous results especially for k=3.

Findings

New upper bound for sum of error terms in additive divisor problem

Improved bounds for k=3 when H ≥ N^{1/2}

Extension of results to general k ≥ 3

Abstract

We obtain a new upper bound for $\sum_{h\le H}\Delta_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $\Delta_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the divisor function generated by $\zeta(s)^k$. When $k=3$ the result improves, for $H\ge N^{1/2}$, the bound given in the recent work \cite{[1]} of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$.