Laplace convolutions of weighted averages of arithmetical functions

Abstract

Let $G(g;x):={\sum }_{n\le x}g(n)$ be the summatory function of an arithmetical function $g(n)$. In this paper, we prove that we can write weighted averages of an arbitrary fixed number $N$ of arithmetical functions $g_j(n),j\in \left\{1,\dots ,N\right\}$ as an integral involving the convolution (in the sense of Laplace) of $G_j(x),j\in \left\{1,\dots ,N\right\}$. Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.