Asymptotic uncorrelations between functions with squarefull kernel and functions of invariant average

Abstract

In 1986, Ivi\'c and Tenenbaum introduced arithmetic functions with squarefull kernel, which are also called $s$-functions. Later, Erd\H{o}s and Ivi\'c gave an asymptotic estimate on the shifted convolution sums of $s$-functions. Recently, Bergelson and Richter studied the orbits along the prime Omega function in a uniquely ergodic topological dynamical system and established a new dynamical generalization of the prime number theorem (PNT). These orbits can be viewed as functions of invariant average under multiplications. In this paper, we show that both $s$-functions and their shifted convolutions are asymptotically uncorrelated to the orbits along the prime Omega function in a uniquely ergodic system. As a consequence, we obtain a refinement of the PNT via the local distribution of $s$-functions. Furthermore, several variants of these results are established as well.