The digamma function, Euler–Lehmer constants and their p-adic counterparts

Abstract

Abstract. The goal of this article is twofold. We first extend a result of Murty and Saradha [7] related to the digamma function at rational arguments. Further, we extend another result of the same authors [8] about the nature of p-adic Euler-Lehmer constants. IntroductionFor a real number x = 0, −1, · · · , the digamma function ψ(x) is the logarithmic derivative of the gamma function defined bywhere γ is Euler's constant. Just like the case of the gamma function, the nature of the values of the digamma function at algebraic or even rational arguments is shrouded in mystery.In the rather difficult subject of irrationality or transcendence, sometimes it is more pragmatic to look at a family of special values as opposed to a single specific value and derive something meaningful. An apt instance here is the result of Rivoal [10] about irrationality of infinitude of odd zeta values as opposed to that of a single specific odd zeta value.In this context, Murty and Saradha [7] in a recent work have made some breakthroughs about transcendence of a certain family of digamma values. In particular, they proved the following. Theorem 1.1 (Murty and Saradha). For any positive integer n > 1, at most one of the φ(n) + 1 numbers in the setis algebraic.In this article, we extend their result and prove the following.2010 Mathematics Subject Classification: 11J91.

On $p$-adic Euler constants

“…This theorem was proved by Murty and Saradha in [18] for F being a prime. Our setting also extends the results of Chatterjee and Gun, see [5].…”

“…Then the quotient group (1 + pO K )/(1 + p m O K ) is finite for all natural numbers m, see [12], page 47. This was also worked out in [5].…”

A note on a variant of a conjecture of Rohrlich

“…In fact, the above propositions are based on the norm idea that was done by Chatterjee and Gun in [6, Proposition 4.1].…”

Shifted Euler constants and a generalization of Euler-Stieltjes constants

“…For more results related to arithmetic nature of γ(r, q) and its generalization see [21,22]. For the work done on p-adic version of the Euler-Lehmer constants γ(r, q) see [12] and [18].…”