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 $H = X^{ 8/33 + \varepsilon}$.
Categories: math.NT