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arXiv · 2401.07531

Laplace convolutions of weighted averages of arithmetical functions

Marco Cantarini·Alessandro Gambini·Alessandro Zaccagnini
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Abstract

Let $G(g;x):=\sum_{n\leq x}g(n)$ 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.

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Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini. 2024-01-15. Laplace convolutions of weighted averages of arithmetical functions. https://arxiv.org/abs/2401.07531

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