Special values of derivatives of $L$-series and generalized Stieltjes constants

Abstract

The connection between derivatives of L(s, f ) for periodic arithmetical functions f at s = 1 and generalized Stieltjes constants has been noted earlier. In this paper, we utilize this link to throw light on the arithmetic nature of L ′ (1, f ) and certain Stieltjes constants. In particular, if p is an odd prime greater than 7, then we deduce the transcendence of at least (p − 7)/2 of the generalized Stieltjes constants, {γ1(a, p) : 1 ≤ a < p}, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath [8].

On $p$-adic Euler constants

“…Now, from [17], Lemma 2.1 we have L(k, g) = 0 for positive integers k ≡ 0 mod(p−1) and consequently g ≡ 0 (see [1], Theorem 11.3). This implies g ≡ 0.…”