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Laplace convolutions of weighted averages of arithmetical functions
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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.
Walter de Gruyter GmbH
Title: Laplace convolutions of weighted averages of arithmetical functions
Description:
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.
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