Abstract
Given an arithmetical function f : ℕ → {−1, 1}, consider the function h f (n) = ∑ n − 1 ν = 1 f(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 q ν is a sequence of positive integers with a corresponding sequence of primitive real characters χ ν (mod q ν ) such that L(s, χ ν ) has a Siegel zero β ν = 1 − 1/(η ν log q ν ), η ν > exp e30, then there exist a positive constant c and a function ε(n) tending to 0 as n → ∞ and such that |h λ (n)|/n ≤ c/ log log η ν + ε(n) uniformly for all n ∈ [q 10 ν , q (log log ην)/3 ν ] such that (n, q ν ) = 1.
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Research supported in part by a grant from NSERC.
Research supported by the Hungarian and Vietnamese TET 10-1-2011-0645.
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De Koninck, JM., Germán, L. & Kátai, I. On the convolution of the Liouville function under the existence of Siegel zeros. Lith Math J 55, 331–342 (2015). https://doi.org/10.1007/s10986-015-9284-x
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DOI: https://doi.org/10.1007/s10986-015-9284-x