Special values of derivatives of $L$-series and generalized Stieltjes constants
Volume 184 / 2018
Acta Arithmetica 184 (2018), 127-138
MSC: Primary 11M41.
DOI: 10.4064/aa170615-13-3
Published online: 7 May 2018
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, $\{ \gamma_1(a,p) : 1 \leq a \lt p \}$, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath (2009).
