On the coefficients of the Baez-Duarte criterion for the Riemann hypothesis and their extensions

TL;DR

This paper explores the properties and extensions of the coefficients in the Baez-Duarte criterion for the Riemann hypothesis, linking them to other constants and generalizing related representations.

Contribution

It introduces new analytic properties and generalizations of the coefficients in the Baez-Duarte criterion, connecting them to various special functions and constants.

Findings

Extended the representation of the reciprocal of the zeta function to Hurwitz zeta and other functions.

Related the coefficients to known constants like Stieltjes constants.

Generalized the Maslanka hypergeometric-like representation for the zeta function.

Abstract

We present analytic properties and extensions of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present generalizations of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the Maslanka hypergeometric-like representation for the zeta function in several ways.