Primes represented by positive definite binary quadratic forms

TL;DR

This paper establishes upper bounds on the number of primes represented by positive definite binary quadratic forms, improving understanding of their distribution within certain ranges, under various hypotheses.

Contribution

It provides new upper bounds for primes represented by quadratic forms, valid unconditionally, under GRH, and for most discriminants, with optimal ranges of x.

Findings

Upper bounds for prime representation within a range of x

Conditional bounds assuming GRH

Results hold for almost all discriminants

Abstract

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH), and for almost all discriminants. The key feature of these estimates is that they hold whenever $x$ exceeds a small power of $D$ and, in some cases, this range of $x$ is essentially best possible. In particular, if $f$ is reduced then this optimal range of $x$ is achieved for almost all discriminants or by assuming GRH. We also exhibit an upper bound for the number of primes represented by $f$ in a short interval and a lower bound for the number of small integers represented by $f$ which have few prime factors.