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 $x$, and any distinct integers $h_1,\dots ,h_k,h_1^{\prime },\dots ,h_{\ell }^{\prime }$, we establish an asymptotic formula for $$\sum _{n\le x}\Lambda (n+h_1)\cdots \Lambda (n+h_k)\lambda (n+h_1^{\prime })\cdots \lambda (n+h_{\ell }^{\prime })$$ for any $0\le k\le 2$ and $\ell \ge 0$. 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.

54 pages, no figures. To appear in J. London Math. Soc

References in corpus (4)

• On the Möbius function in all short intervals
• Siegel Zeros and Sarnak's Conjecture
• Averages of the Möbius function on shifted primes
• On the Hardy-Littlewood-Chowla conjecture on average

Cited by in corpus (4)

• On the Hardy-Littlewood-Chowla conjecture on average
• The Chowla conjecture and Landau-Siegel zeroes
• Prime Tuples and Siegel Zeros
• On variants of Chowla's conjecture