  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 $\Lambda$ is the von Mangoldt function, $d_k$ is the
$k^{\operatorname{th}}$ divisor function, $X$ is large and $k \geq l \geq 2$
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 $\Lambda(n) \Lambda(n + h)$
and we obtained better estimates for the error terms at the price of having to
take $H = X^{8/33 + \varepsilon}$.
