Katz–Sarnak density conjecture for low-lying zeros

Prove that, as analytic conductors in a family of L-functions tend to infinity, the distribution of normalized low-lying zeros near s = 1/2 converges to the scaling limits of eigenvalues near 1 in the corresponding classical compact group (unitary, orthogonal, or symplectic) random matrix ensemble.

Background

The paper describes how Katz and Sarnak predict universal behavior of low-lying zeros across families of L-functions, governed by the symmetry type of the associated monodromy group.

They proved these predictions in the function field setting using Deligne’s equidistribution, but the number-field case remains conjectural.

References

The Katz--Sarnak density conjecture predicts that, as the analytic conductors of $L$-functions in a family tend to infinity, the distribution of their normalized low-lying zeros near the critical point $s = 1/2$ converges to the scaling limits of eigenvalues clustered near $1$ in the corresponding random matrix ensemble .

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Subsubsection Katz–Sarnak theory