Fine-scale distribution of roots of quadratic congruences

Abstract

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence ${\mathrm{\mu }}^2\equiv Dmodm$, with $D>0$ square-free and $D\not\equiv 1mod4$. This is achieved by translating the problem to convergence of certain geodesic random line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.