The distribution of spacings between quadratic residues

TL;DR

This paper investigates the distribution of spacings between quadratic residues modulo highly composite, square-free integers, demonstrating that these spacings follow a Poisson process with exponential distribution as the number of prime factors increases.

Contribution

It proves that all correlation functions of spacings between quadratic residues are Poissonian in the limit of many prime factors, revealing a universal statistical behavior.

Findings

Spacings between quadratic residues follow a Poisson distribution.

Nearest neighbor spacings are exponentially distributed.

Correlation functions are Poissonian in the limit.

Abstract

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.