Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$and$\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$, where $\mathrm{\Lambda }$ is the von Mangoldt function, $d_k$ is the $k^{\mathrm{th}}$ divisor function, $X$ is large and $k\ge l\ge 2$ are real numbers. We show that for almost all $h\in [-H,H]$ with $H = (\log X)^{10000 k \log k}$, the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $\mathrm{\Lambda }(n)\mathrm{\Lambda }(n+h)$ and we obtained better estimates for the error terms at the price of having to take $H=X^{8\mathrm{/}33+\epsilon }$.