Some asymptotic formulae involving Cohen-Ramanujan expansions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916486879117312 |
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| author | Chandran, Arya K, Vishnu Namboothiri |
| author_facet | Chandran, Arya K, Vishnu Namboothiri |
| contents | Cohen-Ramanujan sum, denoted by $c_r^s(n)$, is an exponential sum similar to the Ramanujan sum $c_r(n):=\sum\limits_{\substack{h=1\\{(h,r)=1}}}^{r}e^{\frac{2πi n h}{r}}$. An arithmetical function $f$ is said to admit a Cohen-Ramanujan expansion
$ f(n):=\sum\limits_{r}\widehat{f}(r)c_r^s(n)$ if the series on the right hand side converges for suitable complex numbers $\widehat{f}(r)$. Given two arithmetical functions $f$ and $g$ with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for the sum $\sum\limits_{\substack{n\leq N}}f(n)g(n+h)$ where $h$ is a fixed non negative integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11890 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some asymptotic formulae involving Cohen-Ramanujan expansions Chandran, Arya K, Vishnu Namboothiri Number Theory 11A25, 11L03, 11N05, 11N37 Cohen-Ramanujan sum, denoted by $c_r^s(n)$, is an exponential sum similar to the Ramanujan sum $c_r(n):=\sum\limits_{\substack{h=1\\{(h,r)=1}}}^{r}e^{\frac{2πi n h}{r}}$. An arithmetical function $f$ is said to admit a Cohen-Ramanujan expansion $ f(n):=\sum\limits_{r}\widehat{f}(r)c_r^s(n)$ if the series on the right hand side converges for suitable complex numbers $\widehat{f}(r)$. Given two arithmetical functions $f$ and $g$ with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for the sum $\sum\limits_{\substack{n\leq N}}f(n)g(n+h)$ where $h$ is a fixed non negative integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently. |
| title | Some asymptotic formulae involving Cohen-Ramanujan expansions |
| topic | Number Theory 11A25, 11L03, 11N05, 11N37 |
| url | https://arxiv.org/abs/2411.11890 |