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arXiv · 2404.07690

Transcendental nature of $p$-adic digamma values

Tapas Chatterjee·Sonam Garg
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Abstract

For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $\psi_p(r/p)+ \gamma_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $\psi_p(r/p^n)+ \gamma_p$, 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 $\psi_p(r/pq)+ \gamma_p$, where $p, q$ are distinct primes.

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Tapas Chatterjee, Sonam Garg. 2024-04-11. Transcendental nature of $p$-adic digamma values. https://arxiv.org/abs/2404.07690

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