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

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 ${\sum }_{\rho }\frac{1}{\mathrm{\mid }{\zeta }^{\mathrm{\prime }}(\rho ){\mathrm{\mid }}^2}\ll T^{3\mathrm{/}2-\delta }$, for some $\delta >0$, the value of this constant coincides with the lower bound given by Burnol.