Lemke Oliver and Soundararajan bias for consecutive sums of two squares

TL;DR

This paper investigates biases in the distribution of consecutive sums of two squares in arithmetic progressions, proposing a heuristic model inspired by Hardy--Littlewood conjectures and supporting it with theoretical results.

Contribution

It introduces a heuristic model for sums of two squares in arithmetic progressions and proves related average results for Hardy--Littlewood constants.

Findings

Heuristic model explains biases in sums of two squares data.

Proved results on averages of Hardy--Littlewood constants.

Supports the model with experimental and theoretical evidence.

Abstract

In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several results related to averages of the Hardy--Littlewood constant in the context of sums of two squares.