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On the multiplicative pair correlations of sums of two squares

Jouni Parkkonen, Frédéric Paulin

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.

On the multiplicative pair correlations of sums of two squares

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.
Paper Structure (5 sections, 3 theorems, 135 equations)

This paper contains 5 sections, 3 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Parametrizing the pair correlation data
  3. Uniformisation of the empirical distribution
  4. The main result and its proof
  5. Experiments for bigger scalings

Key Result

Theorem 1.1

Assume that we have $D_K\equiv 0\bmod 4$. Let $\alpha\in\;]0, \frac{1}{2}[$ and $\beta\in\;]0,1+ \frac{\alpha}{2}[\,$. As $N\rightarrow+\infty$, the empirical pair correlation measures ${\cal R}_N^{\alpha,\beta}$ converge, for the weak-star convergence of measures on ${\mathbb R}$, to the asymptotic

Theorems & Definitions (3)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 4.1