On the general additive divisor problem

We obtain a new upper bound for the sum $\sum_{h\le H}\Delta_k(N,h)$ when $1\le H\le N$, $k\in\mathbb N$, $k\ge3$, 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 a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$.