Laplace convolutions of weighted averages of arithmetical functions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914642933055488 |
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| author | Cantarini, Marco Gambini, Alessandro Zaccagnini, Alessandro |
| author_facet | Cantarini, Marco Gambini, Alessandro Zaccagnini, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07531 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Laplace convolutions of weighted averages of arithmetical functions Cantarini, Marco Gambini, Alessandro Zaccagnini, Alessandro Number Theory 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. |
| title | Laplace convolutions of weighted averages of arithmetical functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.07531 |