Title: A criterion related to the Riemann Hypothesis,
Abstract: A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann
Hypothesis is played by the distance \[
d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty\left|1-\zeta
A_N\left(\frac{1}{2}+it\right)\right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] where
the infimum is over all Dirichlet polynomials
$$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$ of length $N$. In this paper we
investigate $d_N^2$ under the assumption that the Riemann zeta function has
four non-trivial zeros off the critical line. Thus we obtain a criterion for
the non validity of the Riemann Hypothesis.