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

1. Problem Statement

The paper addresses a specific counting problem in analytic number theory with applications to low-dimensional topology. The central question is to determine the asymptotic behavior of the number of $SL_2(\mathbb{Z})$-equivalence classes of integral, positive definite binary quadratic forms whose discriminant is of the form $\Delta = 1 - 4p$, where $p$ is a prime number, and $|\Delta| \leq X$ (equivalently $p \leq X$).

Let $H(D)$ denote the number of such equivalence classes for a discriminant $D$. The authors seek an asymptotic formula for the sum:
$$\sum_{p \leq X} H(1 - 4p)$$
This problem arises from a connection established in prior work ([10]) between these quadratic forms and simple $(4a+1)$-knots of genus 1. Specifically, the classification of these knots is linked to the equivalence classes of binary quadratic forms. The authors aim to quantify the "dominant" term in the distribution of these knots when the Alexander polynomial parameters are restricted to primes.

2. Methodology

The authors employ a combination of analytic number theory techniques, specifically the analytic class number formula, bilinear character sums, and random Euler product models.

A. Reduction to Hurwitz Class Numbers and L-functions

The authors first relate the count of all binary quadratic forms (including imprimitive ones) to the Hurwitz class number $Hur(D)$. They utilize the identity:
$$H(D) = \sum_{d^2 | D} h(D/d^2)$$
where $h(D)$ is the classical class number of primitive forms. Using the analytic class number formula $h(D) = \frac{1}{\pi} L(1, \chi_D) |D|^{1/2}$, the problem is transformed into estimating a weighted sum of Dirichlet L-values:
$$Q(X) = \sum_{p \leq X} Hur(1-4p) = \sum_{d \geq 1} \sum_{\substack{p \leq X \\ d^2 | 1-4p}} \frac{1}{\pi d} L\left(1, \chi_{(1-4p)/d^2}\right) \left| \frac{1-4p}{d^2} \right|^{1/2}$$
This reduces the problem to bounding the quantity $T_d(X)$, a weighted sum of L-values over primes $p$ satisfying a congruence condition $d^2 | 1-4p$.

B. Bilinear Character Sums and Siegel-Walfisz

To estimate $T_d(X)$, the authors expand the L-function $L(1, \chi)$ as a series involving the Kronecker symbol:
$$L(1, \chi) = \sum_{n \geq 1} \frac{1}{n} \left( \frac{D}{n} \right)$$
The core technical challenge is bounding the resulting bilinear character sum:
$$\sum_{p \leq X} \sum_{n} \frac{1}{n} \left( \frac{(1-4p)/d^2}{n} \right)$$
The authors split the summation over $n$ into three ranges:

1. Small $n$ ($n \leq (\log X)^B$): They apply the Siegel-Walfisz theorem to estimate the sum over primes $p$. This requires calculating the average of the Kronecker symbol over the arithmetic progression defined by $d^2 | 1-4p$.
2. Medium $n$ ($( \log X)^B < n < X^{1/2+\epsilon}$): They apply unrestricted bilinear character sum bounds (a variant of the Polya-Vinogradov inequality, Lemma 4.2) to show this range contributes only to the error term.
3. Large $n$ ($n \geq X^{1/2+\epsilon}$): This tail is shown to be negligible.

C. Random Euler Product Model

To explain the appearance of Artin's constant in the final asymptotic, the authors construct a probabilistic model. They treat the local factors of the "Zagier L-function" (associated with Hurwitz class numbers) as independent random variables.

• They analyze the distribution of the local L-factors $L_{D,p}(s)$ when $D$ ranges over all integers versus when $D$ is restricted to the form $1-4p$.
• They demonstrate that restricting $D$ to $1-4p$ alters the probability distribution of the local factors (specifically, the density of $D$ being a quadratic residue modulo $\ell$ changes).
• This probabilistic shift results in a global factor of $\prod_{\ell} (1 - \frac{1}{\ell(\ell-1)})$, which is exactly Artin's constant ($C_{Art}$).

3. Key Contributions

1. Exact Asymptotic Formula: The paper provides the first exact asymptotic formula for the sum of Hurwitz class numbers with discriminants $1-4p$, including the explicit constant. Previous work (Friedlander-Iwaniec, Nagoshi) handled discriminants like $-4p$ or $-p$, but the form $1-4p$ introduces specific complications regarding non-primitive forms and residue distributions.
2. Resolution of the "Primitivity" Discrepancy: The authors clarify why restricting to primes in the discriminant form $1-4p$ changes the average size of the class numbers by a factor of $C_{Art}$, whereas similar restrictions in other contexts (like $H(-p)$) do not. This is due to the specific congruence properties of numbers of the form $1-4p$ modulo primes.
3. Random Model for Zagier L-functions: They provide a novel random Euler product model specifically for Zagier L-functions (which govern Hurwitz class numbers), extending the standard application of random models from classical Dirichlet L-functions.
4. Topological Application: They translate the number-theoretic result into a topological corollary regarding the enumeration of simple $(4a+1)$-knots and Seifert matrices with specific Alexander polynomials.

4. Main Results

The primary result is Theorem 1.1:
For $X \geq 2$, the number of $SL_2(\mathbb{Z})$-classes of positive definite binary quadratic forms with discriminant $1-4p$ ($p$ prime) and bounded by $X$ satisfies:
$$\sum_{p \leq X} H(1 - 4p) = C_{Art} \cdot \frac{2\pi}{9} \cdot \frac{X^{3/2}}{\log X} + O\left( \frac{X^{3/2}}{(\log X)^2} \right)$$
where $C_{Art} = \prod_{\ell \geq 2} \left( 1 - \frac{1}{\ell(\ell-1)} \right)$ is Artin's constant.

A more precise version (Theorem 5.3) offers an error term of $O(X^{3/2}(\log X)^{-A})$ for any $A > 1$.

Corollary 1.2 (Topological): The number of $S$-equivalence classes of $2 \times 2$ Seifert matrices $P$ with Alexander polynomial $\det(tP - P^T) = pt^2 + (1-2p)t + p$ for prime $p \in [1, X]$ is asymptotic to $C_{Art} \frac{4\pi}{9} \frac{X^{3/2}}{\log X}$.

5. Significance

• Refinement of Heuristics: The paper corrects a naive heuristic assumption from previous literature. It shows that while the average behavior of class numbers $H(1-4m)$ for all integers $m$ is $\sim \frac{2\pi}{9} X^{3/2}$, restricting $m$ to primes in the specific form $1-4p$ introduces a multiplicative factor of $C_{Art}$. This highlights the subtle interplay between the distribution of primes and quadratic residues.
• Knot Theory: The result provides a quantitative foundation for the study of high-dimensional knots. It confirms that knots with Alexander polynomials of the form $\Delta_p(t)$ (where $p$ is prime) dominate the total count of genus 1 simple knots, and it quantifies this dominance precisely.
• Methodological Innovation: The successful application of the Siegel-Walfisz theorem combined with bilinear sums to handle the specific arithmetic structure of $1-4p$ (which differs from standard prime discriminants) demonstrates a robust technique for similar counting problems involving non-fundamental discriminants.
• Probabilistic Number Theory: The derivation of Artin's constant via a random Euler product model for Zagier L-functions offers a new perspective on the statistical distribution of class numbers, suggesting that such probabilistic models can be extended beyond standard L-functions.