Computation of the derivative of the Hurwitz zeta-function and the higher Kinkelin constants

TL;DR

This paper computes derivatives of the Hurwitz zeta function at negative integers using Glaisher-Kinkelin-Bendersky constants and revisits historical methods, proposing a renaming of the constants.

Contribution

It introduces a numerical approach to evaluate Hurwitz zeta derivatives via GKB constants and highlights historical precedence for these methods.

Findings

Numerical values for Hurwitz zeta derivatives at negative integers.

Historical insight into Glaisher's and Bendersky's methods.

Proposal to rename GKB constants to GKBJ.

Abstract

I use the numerical values of the generalised Glaisher-Kinkelin-Bendersky (GKB) constants to give numerical values for the derivatives of the Hurwitz zeta function at negative integers, rather than the other way round. I point out that both Glaisher's numerical approach and Bendersky's recursion for the generalised gamma function were anticipated by Jeffery in 1862 who gave the value of the second constant as an example. I therefore propose that GKB become GKBJ.