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

Abstract

In this paper 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 (reciprocal) logarithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of the second kind. In addition, two interesting series with rational terms are given for Euler's constant and the constant ln(2*pi), and yet another generalization of Euler's constant is proposed and various formulas for the calculation of these constants are obtained. Finally, in the paper, we mention that almost all the constants considered in this work admit simple representations via the Ramanujan summation.

References in corpus (3)

• Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to ${\pi }^{-1}$
• Expansions of generalized Euler's constants into the series of polynomials in ${\pi }^{-2}$ and into the formal enveloping series with rational coefficients only
• Some possible approaches to the Riemann Hypothesis via the Li/Keiper constants