Arithmetic Biases for Binary Quadratic Forms

The prime number theorem for arithmetic progressions tells us that there are asymptotically as many primes congruent to $1 \\bmod 4$ as there are congruent to $3 \\bmod 4$. That being said, Chebyshev noticed that (numerically) there almost always seems to be slightly more primes congruent to $3$. This simple fact has a highly non-trivial explanation. Rubinstein and Sarnak proved that the assumption of some natural (yet still unproven) conjectures, there is a way to prove that there are more primes congruent to $3$ than congruent to $1$ more than half of the time (in an appropriate sense). \n \n Many other sets of integers demonstrate a bias towards a certain residue class modulo some number $q$. Recently, Gorodetsky showed that the sums of two squares exhibit a Chebyshev-type bias, and that in this case the conjectures one must assume to prove the existence of the bias are weaker. In this thesis, we present two papers which demonstrate some bias in arithmetic progressions for sets of integers that are represented by a given binary quadratic form. \n \n In Chapter 2, 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 are proven, and the paper finishes with some numerical data that is illustrative of the generic data for any quadratic form. \n \n In Chapter 3, we examine a different kind of bias. We ask for the distribution of pairs of sums of two squares in arithmetic progressions, i.e. how many numbers are the sum of two squares, congruent to $a \\bmod q$, and are such that the next largest sum of two squares is congruent to $b \\bmod q$. We prove that when $q \\equiv 1 \\bmod 4$, we have equidistribution among the $q^2$ possible pairs of residue classes. That being said, there exist bizarre numerical biases, most notably a negative bias towards repetition. The main purpose of the second paper is to provide a conjecture which explains the bias, via a secondary and tertiary term in the associated asymptotic expansion. We then support this conjecture with both numerical and theoretical evidence. The paper contains many partial results in the direction of the conjecture, as well as some theorems on the sums of two squares that are of independent interest. For example, we provide an integral representation for the number of integers not exceeding $x$ which are the sum of two squares. This integral representation is akin to $li(x)$ for primes, in that it has a $O(x^{1/2+\\epsilon})$ error term under the Generalized Riemann Hypothesis.

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