The least prime number represented by a binary quadratic form

Abstract

Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval $[X,2X]$. Moreover, assume that ${\pi }_D(X)$ is the number of primes, which split in $\mathbb{Q}(\sqrt{D})$ with norm in the interval $[X,2X].$ We prove that $$(\frac{{\pi }_D(X)}{\pi (X)}{)}^2\ll \frac{R(X,D)}{h(D)}(1+\frac{h(D)}{\pi (X)}),$$ where $\pi (X)$ is the number of primes in the interval $[X,2X]$ and the implicit constant in $\ll$ is independent of $D$ and $X$.