On the convolution of the Liouville function under the existence of Siegel zeros

Abstract

Given an arithmetical function f : ℕ → {−1, 1}, consider the function hf (n) = ∑ν = 1n − 1f(ν)f(n − ν). Let λ be the Liouville function defined by λ(n) = (−1)Ω(n), where Ω(n) stands for the number of prime divisors of n counting their multiplicity. We prove that if qν is a sequence of positive integers with a corresponding sequence of primitive real characters χν (mod qν ) such that L(s, χν) has a Siegel zero βν = 1 − 1/(ην log qν), ην > exp e30, then there exist a positive constant c and a function e(n) tending to 0 as n → ∞ and such that |hλ(n)|/n ≤ c/ log log ην + e(n) uniformly for all n ∈ [qν10 , qν(log log ην)/3] such that (n, qν) = 1.