Triple correlations of multiplicative functions
Volume 180 / 2017
Acta Arithmetica 180 (2017), 63-88
MSC: Primary 11N37; Secondary 11N60.
DOI: 10.4064/aa8605-4-2017
Published online: 1 August 2017
Abstract
We find an 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 multiplicative functions with modulus less than or equal to $1.$
Moreover, under some assumption on $g_1,g_2,$ we prove that as $x\rightarrow \infty,$ \[\frac{1}{x}\sum_{n\le x}g_1(n+3)g_2(n+2)\mu(n+1)=o(1),\] and assuming the $2$-point Chowla type conjecture we show that as $x\rightarrow \infty,$ \[\frac{1}{x}\sum_{n\le x}g_1(n+3)\mu(n+2)\mu(n+1)=o(1).\]
