Item

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 $$ \Big(\frac{\pi_D(X)}{\pi(X)}\Big)^2 \ll
\frac{R(X,D)}{h(D)}\Big(1+\frac{h(D)}{\pi(X)}\Big), $$ where $\pi(X)$ is the
number of primes in the interval $[X,2X]$ and the implicit constant in $\ll$ is
independent of $D$ and $X$.