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

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 $O(x^{1\mathrm{/}2+\epsilon })$.) 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.