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

Alison Beth Miller; Stanley Yao Xiao

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

Number Theory 2026-02-12 v3

Authors: Alison Beth Miller , Stanley Yao Xiao

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.

Keywords

binary quadratic forms discriminant sl2(z)-equivalence hurwitz class numbers euler product asymptotic formula

Cite

@article{arxiv.2011.06559,
  title  = {On the number of binary quadratic forms having discriminant $1-4p$, $p$ prime},
  author = {Alison Beth Miller and Stanley Yao Xiao},
  journal= {arXiv preprint arXiv:2011.06559},
  year   = {2026}
}

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Version 3 is the final version expected to be the published version

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

Abstract

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