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

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 .