On the Riesz and Baez-Duarte criteria for the Riemann Hypothesis

TL;DR

This paper explores the mathematical relationship between two criteria, Riesz and Baez-Duarte, for the Riemann Hypothesis, linking their key functions and sequences to deepen understanding of this famous conjecture.

Contribution

It establishes explicit relations between the Riesz function R(x) and the Baez-Duarte sequence c_k, showing how each can be derived from the other.

Findings

R(x) can be expressed in terms of c_k

c_k can be obtained from R(x) at integer points

Relations involving c_k, R(x), and their sums are derived

Abstract

We investigate the relation between the Riesz and the B{\'a}ez-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 B{\'a}ez-Duarte formulation. It is shown that $R(x)$ can be expressed by $c_k$, and, vice versa, the sequence $c_k$ can be obtained from the values of $R(x)$ at integer arguments. Also, we give some relations involving $c_k$ and $R(x)$, and value of the alternating sum of $c_k$.