Triple Correlations of Multiplicative Functions

2016/08/09 by Pranendu Darbar, Darbar, Pranendu
Mathematics · #11N37 #11N60 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N37 #msc:11N60

paper · pdf · doi:10.48550/arxiv.1608.02759

23 pages

arxiv created 2016/08/09 · arxiv updated 2016/08/10

Abstract

In this paper, we find asymptotic formula for the following sum with explicit error term: M_x(g₁, g₂, g₃)=(1)/(x)∑_{n≤ x}g₁(F₁(n))g₂(F₂(n))g₃ (F₃(n)), where F₁(x), F₂(x) and F₃(x) are polynomials with integer coefficients and g₁,g₂,g₃ are multilpicative functions with modulus less than or equal to 1. Moreover, under some assumption on g₁,g₂, we prove that as x→ ∞, (1)/(x)∑_{n≤ x}g₁(n+3)g₂(n+2)μ(n+1)=o(1) and assuming 2-point Chowla type conjecture we show that as x→ ∞, (1)/(x)∑_{n≤ x}g₁(n+3)μ(n+2)μ(n+1)=o(1).

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