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 the imaginary quadratic field $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of the 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

where $\pi (X)$ is the number of the primes is the number of primes in the interval [$X,2X$] and the implicit constant in $\ll$ is independent of $D$ and $X$.