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

TL;DR

This paper proves a hybrid version of the Hardy--Littlewood and Chowla conjectures assuming Siegel zeros, extending previous results and providing an asymptotic formula for prime tuples in an infinite sequence.

Contribution

It establishes a broader asymptotic formula for prime tuples under the assumption of Siegel zeros, unifying and extending prior results on Hardy--Littlewood and Chowla conjectures.

Findings

Proves a hybrid prime tuples conjecture assuming Siegel zeros.

Extends the range of validity beyond previous results.

Unifies Hardy--Littlewood and Chowla conjectures under a common framework.

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 $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}\Lambda(n+h_1)\cdots \Lambda(n+h_k)\lambda(n+h_{1}')\cdots \lambda(n+h_{\ell}')$$ for any $0\leq k\leq 2$ and $\ell \geq 0$. Specializing to either $\ell=0$ or $k=0$, 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.