@Ga; @GG]; Prime gaps [@CMS].

From our point of view, our main contribution connects here. So far the only class ${\mathcal{A}}$ used for problems and was some Paley-Wiener space of bandlimited functions. We relax the bandlimited condition by requiring only certain sign conditions on the auxiliary function that match exactly with the very same conditions required by the linear programming bounds for the packing problem (see Section \[prelims\] for a detailed explanation). This relation is what ultimately inspired and allowed us to perform numerical computations to find good test functions for the functionals we derive in Section \[prelims\]. Furthermore, as far as we know, it is the first time this method is used in the zeta-function theory.

Main Results
============

\[thmNstar\] Assuming RH we have $$\begin{aligned}
    N^*(T)\leq ({1.3208}+o(1))N(T).
    \end{aligned}$$ Assuming GRH we have $$\begin{aligned}
    N^*(T)\leq ({1.3155}+o(1))N(T).
    \end{aligned}$$

Montgomery [@M] was the first to show the constant $1.3333...$. This result was later improved to $1.3275$ by Cheer and Goldston [@CG]. Assuming the generalized Riemann Hypothesis (GRH), Goldston, Gonek, Özlük and Snyder [@GGOS] improved it to $1.3262$. To the best of our knowledge, our bounds are the current best.

Theorem \[thmNstar\] has an important application to estimating the quantity of simple zeros 