Determination of the Stieltjes constants at rational arguments

TL;DR

This paper reviews historical and recent results on the evaluation of Stieltjes constants at rational arguments, highlighting prior work and providing a simplified expression for these constants.

Contribution

It clarifies the historical development of evaluating Stieltjes constants at rational points and offers a simpler derivation of known results from Chakraborty et al. (2009).

Findings

Deninger determined the first generalized Stieltjes constant at rational arguments in 1984.

Chakraborty, Kanemitsu, and Kuzumaki extended these results in 2009.

The paper presents a simplified expression for these constants based on earlier work.

Abstract

The Stieltjes constants have attracted considerable attention in recent years and a number of authors, including the present one, have considered various ways in which these constants may be evaluated. The primary purpose of this paper is to belatedly highlight the fact that Deninger actually ascertained the first generalised Stieltjes constant at rational arguments as long ago as 1984 and that all of the higher constants (at rational arguments) were determined in principle by Chakraborty, Kanemitsu and Kuzumaki in 2009. Equivalent results were obtained by Musser in his 2011 thesis. The authors of the above papers simply referred to the constants as the Laurent coefficients which explains why various electronic searches conducted by this author for Stieltjes constants did not readily highlight these particular sources. In this paper the author has employed a slightly different argument to obtain a simpler expression for the results originally derived by Chakraborty et al. in 2009.