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

TL;DR

This paper derives an asymptotic formula for counting classes of positive definite binary quadratic forms with discriminant linked to prime numbers, and models the distribution of Hurwitz class numbers.

Contribution

It provides a new asymptotic count for quadratic forms with discriminant 1-4p and introduces a probabilistic model for Hurwitz class number distribution.

Findings

Asymptotic formula for quadratic form classes with discriminant 1-4p

Random Euler product model for Hurwitz class numbers

Supports the model with derived asymptotic results

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.