Laplace convolutions of weighted averages of arithmetical functions

Abstract

Abstract Let G ⁢ ( g ; x ) := ∑ n ≤ x g ⁢ ( n ) {G(g;x):=\sum_{n\leq x}g(n)} be the summatory function of an arithmetical function g ⁢ ( n ) {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 ∈ { 1 , … , N } {g_{j}(n),\,j\in\{1,\dots,N\}} as an integral involving the convolution (in the sense of Laplace) of G j ⁢ ( x ) {G_{j}(x)} , j ∈ { 1 , … , N } {j\in\{1,\dots,N\}} . 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.