On the general additive divisor problem

We obtain a new upper bound for $\sum_{h\le H}Δ_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $Δ_k(N,h)$ 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 $ζ(s)^k$. 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 $k=3$.

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