Prime values of $f(a,b^2)$ and $f(a,p^2)$, $f$ quadratic

TL;DR

This paper establishes an asymptotic count for primes represented by quadratic forms of the type f(a,b^2) and f(a,p^2), refining previous results on primes of specific quadratic forms.

Contribution

It provides a new asymptotic formula for primes of the form f(a,b^2) and f(a,p^2), extending earlier work on primes represented by quadratic forms.

Findings

Asymptotic formula for primes of the form f(a,b^2)

Asymptotic formula for primes of the form f(a,p^2)

Refinement of previous results on quadratic form primes

Abstract

We prove an asymptotic formula for primes of the shape $f(a,b^2)$ with $a,b$ integers and of the shape $f(a,p^2)$ with $p$ prime. Here $f$ is a binary quadratic form with integer coefficients, irreducible over $\mathbb{Q}$ and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form $x^2 + y^4$ and Heath-Brown and Li on primes of the form $a^2 + p^4$, as well as earlier work of the author with Lam and Schindler on primes of the form $f(a,p)$ with $f$ a positive definite form.