On an analytic estimate in the theory of the Riemann zeta function and a theorem of Báez-Duarte.

Abstract

On the Riemann hypothesis we establish a uniform upper estimate for zeta(s)/zeta (s + A), 0 < or = A, on the critical line. We use this to give a purely complex-analytic variant of Báez-Duarte's proof of a strengthened Nyman-Beurling criterion for the validity of the Riemann Hypothesis. We investigate function-theoretically some of the functions defined by Báez-Duarte in his study and we show that their square-integrability is, in itself, an equivalent formulation of the Riemann Hypothesis. We conclude with a third equivalent formulation which resembles a "causality" statement.