Equivalence of Riesz and Baez-Duarte criterion for the Riemann Hypothesis

TL;DR

This paper explores the mathematical equivalence between two criteria, Riesz and Baez-Duarte, for the Riemann Hypothesis, by establishing explicit relations between their defining functions and sequences.

Contribution

It demonstrates the explicit relationship between the Riesz function and Baez-Duarte sequence, showing their equivalence in the context of the Riemann Hypothesis.

Findings

R(x) can be expressed in terms of c_k

c_k can be derived from R(x) at integers

Relations involving c_k and R(x) are established

Abstract

We investigate the relation between the Riesz and the Baez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function $R(x)$ appearing in the Riesz criterion and the sequence $c_k$ appearing in the Baez-Duarte formulation. It is shown that $R(x)$ can expressed by $c_k$ and vice versa the sequence $c_k$ can be obtained from the values of $R(x)$ at integer arguments. We give also some relations involving $c_k$ and $R(x)$, in particular value of the alternating sum of $c_k$.