Transcendental nature of $p$-adic digamma values
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909166791032832 |
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| author | Chatterjee, Tapas Garg, Sonam |
| author_facet | Chatterjee, Tapas Garg, Sonam |
| contents | For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $ψ_p(r/p)+ γ_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $ψ_p(r/p^n)+ γ_p$, for any integer $n>1$. In this article, we generalize their results for distinct prime powers and explore the transcendental nature of the $p$-adic digamma values, with at most one exception. Further, we investigate the multiplicative independence of cyclotomic numbers satisfying certain conditions. Using this, we prove the transcendental nature of $p$-adic digamma values corresponding to $ψ_p(r/pq)+ γ_p$, where $p, q$ are distinct primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transcendental nature of $p$-adic digamma values Chatterjee, Tapas Garg, Sonam Number Theory For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $ψ_p(r/p)+ γ_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $ψ_p(r/p^n)+ γ_p$, for any integer $n>1$. In this article, we generalize their results for distinct prime powers and explore the transcendental nature of the $p$-adic digamma values, with at most one exception. Further, we investigate the multiplicative independence of cyclotomic numbers satisfying certain conditions. Using this, we prove the transcendental nature of $p$-adic digamma values corresponding to $ψ_p(r/pq)+ γ_p$, where $p, q$ are distinct primes. |
| title | Transcendental nature of $p$-adic digamma values |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.07690 |