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

Abstract

<p>We study the problem of obtaining asymptotic formulas for the sums ∑ XX is large and k≥l≥2 k≥l≥2

are integers. We show that for almost all h∈[−H,H] h∈[−H,H]

with H=(logX) 10000klogk H=(log⁡X)10000klog⁡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) 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 H=X8/33+ε

.<br /></p>