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 , … , ℎ 𝑘 , ℎ ′ 1 , … , ℎ ′ 𝓁 , we establish an asymptotic formula forfor any 0 ⩽ 𝑘 ⩽ 2 and 𝓁 ⩾ 0. Specializing to either 𝓁 = 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.M S C 2 0 2 0 11N37 (primary), 11N36 (secondary) 1Let 𝜆 ∶ ℕ → {−1, +1} denote the Liouville function. We have the following well known conjecture of Chowla [3]:

On the Hardy–Littlewood–Chowla conjecture on average

“…The 'pure' cases 𝑘 = 0 and ℓ = 0 specialise to the original conjectures of Hardy-Littlewood and of Chowla, respectively. We remark that under the hypothetical assumption of infinitely many Siegel zeros, Conjecture 1.1 was recently verified for ℓ ≤ 2 by Tao and the second author [20] (generalising works of Heath-Brown [7] and Chinis [1]). In the current paper, we will be concerned with unconditional results.…”

“…It is natural to consider the following hybrid conjecture, which, following [10] and [20], we call the Hardy-Littlewood-Chowla conjecture.…”