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

TL;DR

This paper investigates asymptotic formulas for divisor and von Mangoldt function correlations over short ranges, showing that for most shifts within a large interval, expected estimates hold with high probability.

Contribution

It extends previous work by establishing asymptotic formulas for divisor and von Mangoldt correlations in short ranges for almost all shifts, with improved error estimates compared to prior results.

Findings

Asymptotic formulas hold for almost all shifts in a large interval.

High probability results for divisor and von Mangoldt correlations.

Improved error bounds compared to previous work.

Abstract

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 $H = X^{8/33 + \varepsilon}$.