Czechoslovak Mathematical Journal, Vol. 71, No. 1, pp. 283-308, 2021


On $p$-adic Euler constants

Abhishek Bharadwaj

Received July 29, 2019.   Published online October 9, 2020.

Abstract:  The goal of this article is to associate a $p$-adic analytic function to the Euler constants $\gamma_p (a, F)$, study the properties of these functions in the neighborhood of $s=1$ and introduce a $p$-adic analogue of the infinite sum $\sum_{n \ge1} f(n) / n$ for an algebraic valued, periodic function $f$. After this, we prove the theorem of Baker, Birch and Wirsing in this setup and discuss irrationality results associated to $p$-adic Euler constants generalising the earlier known results in this direction. Finally, we define and prove certain properties of $p$-adic Euler-Briggs constants analogous to the ones proved by Gun, Saha and Sinha.
Keywords:  $p$-adic Euler-Lehmer constant; linear forms in logarithms
Classification MSC:  11J91
DOI:  10.21136/CMJ.2020.0336-19

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Affiliations:   Abhishek Bharadwaj, Chennai Mathematical Institute, H1 State Industries Promotion Corporation of Tamil Nadu Limited IT Park, Old Mahabalipuram Road, Siruseri, Kellambakkam, 603103 Chennai, India, e-mail: abhitvt@cmi.ac.in


 
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