The Hardy-Littlewood-Chowla conjecture in the presence of a Siegel zero

Abstract

Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy-Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers.., and any distinct integers h(1), ... , h(k), h'(1), ... , h'(l), we establish an asymptotic formula forSigma(n <= x) Lambda(n + h(1)) ... Lambda(n + h(k))lambda(n + h'(1)) ... lambda(n + h'(l))for any 0 <= k <= 2 and l >= 0. Specializing to either l = 0 or.. = 0, we deduce the previously known results on the Hardy-Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to Heath-Brown and Chinis, respectively. The range of validity of our asymptotic formula is wider than in these previous results.