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 x $x$ , and any distinct integers h 1 , ⋯ , h k , h 1 ′ , ⋯ , h ℓ ′ $h_1,\dots ,h_k,h^{\prime }_1,\dots ,h^{\prime }_\ell$ , we establish an asymptotic formula for ∑ n ⩽ x Λ ( n + h 1 ) ⋯ Λ ( n + h k ) λ ( n + h 1 ′ ) ⋯ λ ( n + h ℓ ′ ) \begin{equation*} \hspace*{13pt}\sum _{n\leqslant x}\Lambda (n+h_1)\cdots \Lambda (n+h_k)\lambda (n+h_{1}^{\prime })\cdots \lambda (n+h_{\ell }^{\prime })\hspace*{-13pt} \end{equation*} for any 0 ⩽ k ⩽ 2 $0\leqslant k\leqslant 2$ and ℓ ⩾ 0 $\ell \geqslant 0$ . Specializing to either ℓ = 0 $\ell =0$ or k = 0 $k=0$ , 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.