Sum of Two Squares - Pair Correlation and Distribution in Short Intervals

TL;DR

This paper explores the distribution of integers representable as sums of two squares within short intervals, proposing a conjecture on their pair correlation, and demonstrates that their count follows a Poisson distribution under this assumption.

Contribution

It introduces a new conjecture for the pair correlation of sums of two squares and connects it to their distribution in short intervals, extending the approach to other quadratic forms.

Findings

Distribution of sums of two squares in short intervals is Poissonian under the conjecture.

A novel method for formulating pair correlation conjectures using prime power residue rings.

Numerical evidence supports the proposed conjectures and their implications.

Abstract

In this work we show that based on a conjecture for the pair correlation of integers representable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the number of representable integers in short intervals is consistent with a Poissonian distribution, where "short" means of length comparable to the mean spacing between sums of two squares. In addition we present a method for producing such conjectures through calculations in prime power residue rings and describe how these conjectures, as well as the above stated result, may by generalized to other binary quadratic forms. While producing these pair correlation conjectures we arrive at a surprising result regarding Mertens' formula for primes in arithmetic progressions, and in order to test the validity of the conjectures, we present numericalz computations which support our approach.