An optimal choice of Dirichlet polynomials for the Nyman-Beurling criterion

TL;DR

This paper provides a conditional result relating the Riemann Hypothesis to the Nyman-Beurling criterion, assuming a specific bound on the sum over zeta zeros, and shows the constant matches the known lower bound.

Contribution

It establishes a conditional equivalence between the Riemann Hypothesis and a specific bound on the constant in the Nyman-Beurling criterion, refining previous understanding.

Findings

Assumes Riemann Hypothesis and a bound on sum over zeros

Shows the constant matches Burnol's lower bound under conditions

Provides a conditional link between RH and Nyman-Beurling criterion

Abstract

We give a conditional result on the constant in the B\'aez-Duarte reformulation of the Nyman-Beurling criterion for the Riemann Hypothesis. We show that assuming the Riemann hypothesis and that $\sum_{\rho}\frac{1}{|\zeta'(\rho)|^2}\ll T^{3/2-\delta}$, for some $\delta>0$, the value of this constant coincides with the lower bound given by Burnol.