On the number of binary quadratic forms having discriminant $1-4p$, $p$ prime

Abstract

In this paper we obtain an asymptotic formula for the number of $\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\bZ$ having bounded discriminant $\Delta = 1-4p$, with $p$ a prime. We also give a random Euler product model for the distribution of Hurwitz class numbers, which is supported by our formula.