Two identities involving Cohen-Ramanujan expansions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908695982505984 |
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| author | Chandran, Arya Namboothiri, K Vishnu |
| author_facet | Chandran, Arya Namboothiri, K Vishnu |
| contents | An arithmetical function $f$ is said to admit a \emph{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 $\widehat{f}(r)$. Here $c_r^s(n)$ denotes the Cohen-Ramanujan sum defined by E. Cohen. We deduce here a Cohen-Ramanujan expansion for the Jordan totient function $J_k(n)$. Further, we give an an asymptotic formula for the sum $\sum\limits_{n \leq N} \frac{J_a(n)}{n^a} \frac{J_b(n+h)}{(n+h)^b}$ using the expansion we derive. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two identities involving Cohen-Ramanujan expansions Chandran, Arya Namboothiri, K Vishnu Number Theory 11A25, 11L03, 11N05, 11N37 An arithmetical function $f$ is said to admit a \emph{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 $\widehat{f}(r)$. Here $c_r^s(n)$ denotes the Cohen-Ramanujan sum defined by E. Cohen. We deduce here a Cohen-Ramanujan expansion for the Jordan totient function $J_k(n)$. Further, we give an an asymptotic formula for the sum $\sum\limits_{n \leq N} \frac{J_a(n)}{n^a} \frac{J_b(n+h)}{(n+h)^b}$ using the expansion we derive. |
| title | Two identities involving Cohen-Ramanujan expansions |
| topic | Number Theory 11A25, 11L03, 11N05, 11N37 |
| url | https://arxiv.org/abs/2503.12027 |