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

TL;DR

This paper explores the relationship between derivatives of $L$-series at 1 and generalized Stieltjes constants, revealing their arithmetic nature and proving the transcendence of many such constants under certain conjectures.

Contribution

It establishes a new connection between $L$-series derivatives and Stieltjes constants, and proves their transcendence for many cases assuming a conjecture.

Findings

Transcendence of at least (p-7)/2 generalized Stieltjes constants for primes p > 7

Link between derivatives of $L$-series at 1 and generalized Stieltjes constants

Conditional proof based on a conjecture of Gun, Murty, and Rath

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, $\{ \gamma_1(a,p) : 1 \leq a < p \}$, conditional on a conjecture of S. Gun, M. R. Murty and P. Rath.