On the Hardy-Littlewood-Chowla conjecture on average

There has been recent interest in a hybrid form of the celebrated conjectures of Hardy-Littlewood and of Chowla. We prove that for any k,l >= 1 and distinct integers h(2), ..., h(k), a(1), ...., a(l), we have:Sigma(n for all except o(H) values of h(1) = (log X) (l+epsilon). This improves on the range H >= (log X)(psi (X)) , psi(X) -> infinity, obtained in previous work of the first author. Our results also generalise from the Mobius function mu to arbitrary (non-pretentious) multiplicative functions.