Laplace convolutions of weighted averages of arithmetical functions

Let G (g; x ):= Sigma(n <= 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 is an element of {1 , ..., N } as an integral involving the convolution (in the sense of Laplace) of G(j)(x) , j is an element of{1, ... , 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.