that in these special dimensions the linear programming approach could exactly act as the dual problem. This is what inspired Viazovska [@V; @CKMRV] to follow their program and solve the sphere packing problem in dimensions $8$ and $24$. What is interesting and surprising to us is that the same space ${\mathcal{A}}_{LP}$ can be used (but with a functional different than ${\mathcal{Q}}(f)$) to produce numerical bounds in analytic number theory.

The general strategy to study problems and is based on Montgomery’s function $$\begin{aligned}
F(x,T)=\frac{1}{N(T)}\sum_{0<\gamma, \gamma'\leq T}T^{ix(\gamma-\gamma')}w(\gamma-\gamma'),\end{aligned}$$ where the sum is over pairs of ordinates of zeros (with multiplicity) of $\zeta(s)$ and $w(u)=\frac{4}{4+u^2}$. The first step is to use Fourier inversion to obtain $$\label{explicit_formula}
\displaystyle\sum_{0<\gamma, \gamma'\leq T}g\bigg((\gamma-\gamma')\dfrac{\log T}{2\pi}\bigg)w(\gamma-\gamma')=N(T)\int_{-\infty}^{\infty}\widehat{g}(x)F(x,T){\,\text{\rm d}}x,$$ for suitable functions $g$, and use some known asymptotic estimate for $F(x,T)$ as $T\to\infty$ (which is proven only under RH or GRH). Secondly, after a series of inequalities, we produce a minimization problem over ${\mathcal{A}}_{LP}$ for some functional ${\mathcal{Z}}$. We then approach the problem numerically, using the class of functions used for the sphere packing problem in [@CE] and sum-of-squares/semidefinite programming techniques to optimize over these functions. The same basic strategy can be, in principle, carried out for other functions where we have 