Laplace convolutions of weighted averages of arithmetical functions

AbstractLetG⁢(g;x):=∑n≤xg⁢(n){G(g;x):=\sum_{n\leq x}g(n)}be the summatory function of an arithmetical functiong⁢(n){g(n)}. In this paper, we prove that we can write weighted averages of an arbitrary fixed numberNof arithmetical functionsgj⁢(n),j∈{1,…,N}{g_{j}(n),\,j\in\{1,\dots,N\}}as an integral involving the convolution (in the sense of Laplace) ofGj⁢(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.