Determining optimal test functions for $2$-level densities

Katz and Sarnak conjectured a correspondence between the $n$-level density statistics of zeros from families of $L$-functions with eigenvalues from random matrix ensembles, and in many cases the sums of smooth test functions, whose Fourier transforms are finitely supported over scaled zeros in a family, converge to an integral of the test function against a density $W_{n, G}$ depending on the symmetry $G$ of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of $L$-functions. We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger $n$ and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when $n=2$, minimizing \[ \frac{1}{\Phi(0, 0)} \int_{{\mathbb R}^2} W_{2,G} (x, y) \Phi(x, y) dx dy \] over test functions $\Phi : {\mathbb R}^2 \to [0, \infty)$ with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form $\phi(x) \psi(y)$ for some fixed admissible $\psi(y)$ and $\operatorname{supp}{\widehat \phi} \subseteq [-1, 1]$. Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal $\phi$ for appropriately chosen fixed test function $\psi$. We conclude by discussing further improvements on estimates by the method of iteration.