On the number of binary quadratic forms having discriminant $1-4p$, $p$ prime
Alison Beth Miller, Stanley Yao Xiao
TL;DR
The paper determines an asymptotic count for the number of $ ext{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms with discriminant $1-4p$ for prime $p$, up to a bound $X$. It develops two proofs using the analytic class number formula and, separately, a random Euler product model for Zagier $L$-functions associated to Hurwitz class numbers, revealing that Artin’s constant $C_{\operatorname{Art}}=\prod_{\ell\ge2}(1-1/(\ell(\ell-1)))$ governs the average behavior for discriminants of the form $1-4p$ with $p$ prime. The main result is the precise asymptotic $\sum_{p\le X} H(1-4p)= C_{\operatorname{Art}} \frac{2\pi}{9} \frac{X^{3/2}}{\log X} + O\bigl( \frac{X^{3/2}}{(\log X)^2} \bigr)$, with a corollary translating this into counts of $S$-equivalence classes of $2\times2$ Seifert matrices whose Alexander polynomial equals $p t^2+(1-2p)t+p$ for primes $p\le X$. The work connects number theory with knot theory, clarifying how restricting to discriminants tied to primes affects average class numbers and producing a concrete topological interpretation via Seifert data.
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 $\mathbb{Z}$ having bounded discriminant $Δ= 1-4p$, with $p$ a prime. This extends work of the first named author and has an application in counting simple $(4a+1)$-knots of genus one.