Some asymptotic formulae involving Cohen-Ramanujan expansions
| Autor: | Chandran, Arya, K, Vishnu Namboothiri |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | Cohen-Ramanujan sum, denoted by $c_r^s(n)$, is an exponential sum similar to the Ramanujan sum $c_r(n):=\sum\limits_{\substack{h=1\\{(h,r)=1}}}^{r}e^{\frac{2\pi i n h}{r}}$. 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. Comment: arXiv admin note: substantial text overlap with arXiv:2303.09363 |
| Databáze: | arXiv |
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