Katz–Sarnak density conjecture for families of L-functions

Let $L(s)$ be an $L$-function with nontrivial zeros written as $\rho=\frac12+i\gamma$, let $\tilde{\gamma}=\frac{\log c(L)}{2\pi}\gamma$, and for an even Schwartz function $\phi$ on $\mathbb{R}$ with compactly supported Fourier transform define

$$D(L,\phi):=\sum_\gamma\phi(\tilde{\gamma}).$$

Let $\mathcal{F}$ be a family of $L$-functions, and let $\mathcal{F}_X$ be a finite truncation increasing to $\mathcal{F}$ as $X$ grows. Katz–Sarnak density conjecture. There is a classical group $G$ among $\mathrm{U}$, $\mathrm{SO(even)}$, $\mathrm{SO(odd)}$, $\mathrm{O}$, and $\mathrm{Sp}$ such that

$$\frac{1}{|\mathcal{F}_X|}\sum_{f\in\mathcal{F}_X}D(L,\phi)\xrightarrow[X\to\infty]{}\int_{\mathbb{R}}\phi(x)W_G(x)\,dx,$$

where, for all $x\in\mathbb{R}$,

$$W_{\mathrm{U}}(x)=1,\qquad W_{\mathrm{O}}(x)=1+\frac12\delta_0(x),$$ $$W_{\mathrm{SO(even)}}(x)=1+\frac{\sin 2\pi x}{2\pi x},\qquad W_{\mathrm{SO(odd)}}(x)=1-\frac{\sin 2\pi x}{2\pi x}+\frac12\delta_0(x),$$ $$W_{\mathrm{Sp}}(x)=1-\frac{\sin 2\pi x}{2\pi x},$$

with $\delta_0$ the Dirac distribution at $0$; the family is then said to have type of symmetry $G$. For families over function fields, the type of symmetry is determined by monodromy, but no analogous result is known for number fields, and no case of the full conjecture has been proved.