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

Abstract

We investigate the relation between the Riesz and the B{á}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{á}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 $c_k$.

Partly based on arXiv:math.NT/0607782, most of proofs changed, new figures. Some of the research results have been extracted and various new results added. New conjecture is formulated at the end