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

Abstract

We study the problem of obtaining asymptotic formulas for the sums ${\sum }_{X<n\le 2X}{d}_k(n)d_l(n+h)$ and ${\sum }_{X<n\le 2X}\Lambda (n)d_k(n+h)$, where $\Lambda$ 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 $H=(\log X{)}^{10000k\log k}$, 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 $H=X^{8/33+\epsilon }$.

46 pages; incorporated referee comments and corrected a few additional typos

References in corpus (1)

• Linear correlations of multiplicative functions

Cited by in corpus (2)

• Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges
• Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions