A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants

TL;DR

This paper derives a recurrence relation connecting Li/Keiper constants with Stieltjes constants, introduces formulas expressing Stieltjes constants via derivatives of the Riemann zeta function, and uses Bell polynomials involving eta constants.

Contribution

It presents a novel recurrence relation for Li/Keiper constants based on Stieltjes constants and new formulas linking Stieltjes constants to derivatives of the Riemann zeta function.

Findings

Recurrence relation for Li/Keiper constants in terms of Stieltjes constants

Formula expressing Stieltjes constants via derivatives of the Riemann zeta function

Representation of Stieltjes constants using Bell polynomials with eta constants

Abstract

A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants is derived in this paper. In addition, we also report a formula for the Stieltjes constants in terms of the higher derivatives of the Riemann zeta function. A formula for the Stieltjes constants in terms of the (exponential) complete Bell polynomials containing the eta constants as the arguments is also derived.