Triple Correlations of Multiplicative Functions

TL;DR

This paper derives asymptotic formulas for sums involving multiplicative functions evaluated at polynomial arguments, and investigates their correlations with the Möbius function, under certain conjectural assumptions.

Contribution

It provides explicit asymptotic formulas with error terms for triple correlations of multiplicative functions at polynomial arguments and explores their orthogonality to the Möbius function.

Findings

Asymptotic formula for the sum with explicit error term.

Orthogonality results of multiplicative functions to the Möbius function.

Conditional results based on Chowla-type conjectures.

Abstract

In this paper, we find asymptotic formula for the following sum with explicit error term: \[M_{x}(g_{1}, g_{2}, g_3)=\frac{1}{x}\sum_{n\le x}g_{1}(F_1(n))g_{2}(F_2(n))g_{3} (F_3(n)),\] where $F_1(x), F_2(x)$ and $F_3(x)$ are polynomials with integer coefficients and $g_1,g_2,g_3$ are multilpicative functions with modulus less than or equal to $1.$ Moreover, under some assumption on $g_1,g_2,$ we prove that as $x\rightarrow \infty,$ \[\frac{1}{x}\sum\limits_{n\le x}g_1(n+3)g_2(n+2)\mu(n+1)=o(1)\] and assuming $2$-point Chowla type conjecture we show that as $x\rightarrow \infty,$ \[\frac{1}{x}\sum\limits_{n\le x}g_1(n+3)\mu(n+2)\mu(n+1)=o(1).\]