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Keywords:
$p$-adic Euler-Lehmer constant; linear forms in logarithms
$p$-adic Euler-Lehmer constant; linear forms in logarithms
Summary:
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 $\sum _{n \ge 1} f(n) / n$ 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.
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