On the multiplicative pair correlations of sums of two squares

TL;DR

This paper investigates the pair correlations of the logarithms of quadratic norm form values, revealing diverse asymptotic behaviors including Poissonian and exotic transitions as scaling varies.

Contribution

It introduces the study of pair correlation measures for sums of two squares and describes novel asymptotic behaviors at different scalings.

Findings

Existence of pair correlation measures for quadratic norm form values

Transition from punctual to Poissonian behavior with scaling

Identification of exotic asymptotic behavior at transition phases

Abstract

We study the pair correlations of the logarithms of the integral values of quadratic norm forms at various scalings, proving the existence of pair correlation measures. We describe a surprising set of asymptotic behaviours when the scaling increases, passing from a punctual measure to a Poissonian behaviour through an exotic behaviour at the transition phase.