Absolute convergence of Ramanujan expansions admits coefficients' coexistence of Ramanujan expansions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912239030632448 |
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| author | Coppola, Giovanni |
| author_facet | Coppola, Giovanni |
| contents | In this self-contained short note, we prove that {\it every arithmetic function} $F$ {\it has infinitely many Ramanujan coefficients} $G$ {\it giving an absolutely convergent Ramanujan expansion for $F$}. This is "coefficients' coexistence": the non-uniqueness, once fixed any $F$, of these $G$. They are infinitely many ! |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Absolute convergence of Ramanujan expansions admits coefficients' coexistence of Ramanujan expansions Coppola, Giovanni Number Theory 11N37 In this self-contained short note, we prove that {\it every arithmetic function} $F$ {\it has infinitely many Ramanujan coefficients} $G$ {\it giving an absolutely convergent Ramanujan expansion for $F$}. This is "coefficients' coexistence": the non-uniqueness, once fixed any $F$, of these $G$. They are infinitely many ! |
| title | Absolute convergence of Ramanujan expansions admits coefficients' coexistence of Ramanujan expansions |
| topic | Number Theory 11N37 |
| url | https://arxiv.org/abs/2502.14415 |