On the discrete convolution of the Liouville and Möbius functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915851862540288 |
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| author | Cantarini, Marco Gambini, Alessandro Zaccagnini, Alessandro |
| author_facet | Cantarini, Marco Gambini, Alessandro Zaccagnini, Alessandro |
| contents | In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}λ\left(m_{1}\right)λ\left(m_{2}\right)$, which is a Goldbach-type counting function of representations. In particular, using the general approach introduced in a recent paper \cite{CGZ}, we will give an explicit formula for weighted averages of $S(n)$ with a general weights $f(w)$ that verify suitable conditions. This formula allows us to obtain interesting information about the Dirichlet and power series of $S(n)$ and the discrete convolution with an arbitrary numbers of factors $λ(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10241 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the discrete convolution of the Liouville and Möbius functions Cantarini, Marco Gambini, Alessandro Zaccagnini, Alessandro Number Theory In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}λ\left(m_{1}\right)λ\left(m_{2}\right)$, which is a Goldbach-type counting function of representations. In particular, using the general approach introduced in a recent paper \cite{CGZ}, we will give an explicit formula for weighted averages of $S(n)$ with a general weights $f(w)$ that verify suitable conditions. This formula allows us to obtain interesting information about the Dirichlet and power series of $S(n)$ and the discrete convolution with an arbitrary numbers of factors $λ(n)$. |
| title | On the discrete convolution of the Liouville and Möbius functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.10241 |