Fine-scale distribution of roots of quadratic congruences

Jens Marklof; Matthew Welsh

doi:10.1215/00127094-2022-0081

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arXiv · 2105.02854

Fine-scale distribution of roots of quadratic congruences

Jens Marklof·Matthew Welsh

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Abstract

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence $\mu^2 \equiv D \bmod m$, with $D > 0$ square-free and $D\not\equiv 1 \bmod 4$. 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.

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Jens Marklof, Matthew Welsh. 2021-05-06. Fine-scale distribution of roots of quadratic congruences. https://doi.org/10.1215/00127094-2022-0081

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