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

## TL;DR

This paper introduces new rational series involving Gregory coefficients for calculating Stieltjes and Euler's constants, providing novel formulas and representations that connect these constants with classical number sequences.

## Contribution

It presents new series for Stieltjes constants using Gregory coefficients, and offers generalized formulas and simple Ramanujan summation representations for these constants.

## Key findings

- New series for Stieltjes constants involving Gregory coefficients
- Explicit formulas for Euler's constant and ln(2*pi)
- Representation of constants via Ramanujan summation

## 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.