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 ∑_(X<n≤2X)d_k(n)dl(n+h) and ∑_(X<n≤2X)Λ(n)d_k(n+h), where Λ is the von Mangoldt function, dk is the kth divisor function, X is large and k≥l≥2are real numbers. We show that for almost all h∈[−H,H] with H=(logX)^(10000k log k), the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of Λ(n)Λ(n+h) and we obtained better estimates for the error terms at the price of having to take H=X^(8/33+ε).