Triple correlations of multiplicative functions

Pranendu Darbar¹

¹ Institute of Mathematical Sciences CIT Campus, Taramani Chennai 600113, India and Homi Bhabha National Institute Training School Complex Anushakti Nagar Mumbai 400094, India

Acta Arithmetica, Tome 180 (2017) no. 1, pp. 63-88

Résumé

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 $x\rightarrow \infty,$ \[\frac{1}{x}\sum_{n\le x}g_1(n+3)\mu(n+2)\mu(n+1)=o(1).\]

Détail

DOI : 10.4064/aa8605-4-2017

Keywords: asymptotic formula following sum explicit error term frac sum where polynomials integer coefficients multiplicative functions modulus equal nbsp moreover under assumption prove rightarrow infty frac sum assuming point chowla type conjecture rightarrow infty frac sum