Biases Towards the Zero Residue Class for Quadratic Forms in Arithmetic Progressions

TL;DR

This paper investigates biases towards the zero residue class in integers represented by binary quadratic forms, identifying cases where secondary terms cause biases and proposing conjectures for the general behavior.

Contribution

It proves the origin of biases from secondary terms in some cases and formulates conjectures for the general distribution of represented integers.

Findings

Biases often originate from secondary terms in asymptotic expansions.

Numerical data supports the existence of biases even when unproven.

Several new results on the distribution of quadratic form representations are established.

Abstract

We examine a bias towards the zero residue class for the integers represented by binary quadratic forms. In many cases, we are able to prove that the bias comes from a secondary term in the associated asymptotic expansion (unlike Chebyshev's bias, which lives somewhere at the level of $O(x^{1/2+\epsilon})$.) In some other cases, we are unable to prove that a bias exists, even though it is present numerically. We then make a conjecture on the general situation which includes the cases we could not prove. Many interesting results on the distribution of the integers represented by a quadratic form -- some of which are of independent interest -- are proven along the way. The paper concludes with some numerical data that is illustrative of the aforementioned bias.