A note on some constants related to the zeta-function and their relationship with the Gregory coefficients

Abstract

In this article, new series for the first and second Stieltjes constants (also known as generalized Euler’s constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind and the Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler’s constant $$\gamma $$ and the constant $$\ln 2\pi $$ are given, and yet another generalization of Euler’s constant is proposed and various formulas for the calculation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation.