Constancy of an Infinite Cyclotomic Product via Ramanujan Sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917185555791872 |
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| author | Bal, Hartosh Singh |
| author_facet | Bal, Hartosh Singh |
| contents | We show that the infinite product defined by \[ P(z) = -\prod_{n=1}^{\infty} (Φ_n(z))^{-1/n}, \] where \( Φ_n(z) \) is the \( n \)-th cyclotomic polynomial, is constant inside the unit disk. The proof translates a result of Ramanujan on Ramanujan sums, equivalent to the prime number theorem, to the setting of infinite products. We also show that similar identities proved by Ramanujan lead to additional results on infinite cyclotomic products. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constancy of an Infinite Cyclotomic Product via Ramanujan Sums Bal, Hartosh Singh Number Theory Combinatorics 11A25 (Primary) 11R18, 30B10, 11C08 (Secondary) We show that the infinite product defined by \[ P(z) = -\prod_{n=1}^{\infty} (Φ_n(z))^{-1/n}, \] where \( Φ_n(z) \) is the \( n \)-th cyclotomic polynomial, is constant inside the unit disk. The proof translates a result of Ramanujan on Ramanujan sums, equivalent to the prime number theorem, to the setting of infinite products. We also show that similar identities proved by Ramanujan lead to additional results on infinite cyclotomic products. |
| title | Constancy of an Infinite Cyclotomic Product via Ramanujan Sums |
| topic | Number Theory Combinatorics 11A25 (Primary) 11R18, 30B10, 11C08 (Secondary) |
| url | https://arxiv.org/abs/2511.16975 |