On the general additive divisor problem

Abstract

We obtain a new upper bound for ${\sum }_{h\le H}{\Delta }_k(N,h)$ for $1\le H\le N$, $k\in \mathbb{N}$, $k\ge 3$, where ${\Delta }_k(N,h)$ is the (expected) error term in the asymptotic formula for ${\sum }_{N<n\le 2N}{d}_k(n)d_k(n+h)$, 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 $k=3$.

References in corpus (1)

• Averages of shifted convolutions of $d_3(n)$