[Paper Review] On the average distribution of primes represented by binary quadratic forms

Jakob Johann Ditchen|arXiv (Cornell University)|Dec 5, 2013

Analytic Number Theory Research18 references3 citations

TL;DR

This paper establishes Bombieri–Vinogradov and Barban–Davenport–Halberstam-type theorems for primes represented by positive definite binary quadratic forms across long ranges of negative fundamental discriminants. It shows that the least prime represented by such forms is bounded by |q|^{7+ε} for all forms to most discriminants, and by |q|^{3+ε} for most forms to most discriminants, using large sieve inequalities and character sum estimates in ideal class groups.

ABSTRACT

We investigate the average distribution of primes represented by positive definite integral binary quadratic forms, the average being taken over negative fundamental discriminants in long ranges. In particular, we prove corresponding results of Bombieri-Vinogradov type and of Barban-Davenport-Halberstam type, although with shorter ranges than in the original theorems for primes in arithmetic progressions: The results imply that, for all $a>0$, the least prime that can be represented by any given positive definite binary quadratic form of discriminant $q$ is smaller than $|q|^{7+a}$ for all forms to "most" discriminants; moreover, it is even smaller than $|q|^{3+a}$ for "most" forms to "most" discriminants.

Motivation & Objective

• To extend Bombieri–Vinogradov and Barban–Davenport–Halberstam theorems to primes represented by binary quadratic forms, rather than just arithmetic progressions.
• To analyze the average distribution of primes represented by positive definite integral binary quadratic forms over long ranges of negative fundamental discriminants.
• To establish effective upper bounds on the least prime represented by any given form, conditional on the discriminant and class group structure.
• To use large sieve inequalities and character sum estimates in ideal class groups to derive mean-square and maximal error bounds for prime counting functions across form classes.
• To provide quantitative improvements on the best-known upper bounds for the least prime of the form $x^2 + ny^2$, under natural density assumptions.

Proposed method

• Derives a large sieve inequality for complex ideal class group characters, using analytic number theory techniques to control character sums over class groups.
• Applies smooth Bombieri–Vinogradov-type estimates by bounding maximal error terms over all form classes in the class group $\mathcal{K}(q)$ for discriminants $q \in \mathfrak{F}(Q)$.
• Employs real and complex character sum estimates to control the error terms in the prime counting function $\pi(X;q,C)$, leveraging bounds on $L$-functions and class number estimates.
• Establishes a general Barban–Davenport–Halberstam-type result by averaging over both discriminants and form classes, using mean-square error bounds.
• Uses the Lindelöf Hypothesis as a conditional assumption to improve the range of validity and reduce the exponent in the bound on the least prime.
• Applies the Siegel–Walfisz theorem for binary quadratic forms and log-free zero-density estimates to derive unconditional and conditional bounds on the least prime represented.

Experimental results

Research questions

• RQ1 What is the average distribution of primes represented by positive definite binary quadratic forms across long ranges of negative fundamental discriminants?
• RQ2 Can Bombieri–Vinogradov-type theorems be extended to the setting of binary quadratic forms, and with what range of discriminants?
• RQ3 How does the least prime represented by a given binary quadratic form of discriminant $q$ grow in terms of $|q|$?
• RQ4 What is the density of discriminants for which the least prime represented is smaller than $|q|^{3+\varepsilon}$?
• RQ5 Can the assumption that the least prime of the form $x^2 + ny^2$ is achieved with $y=1$ be used to derive average upper bounds on the least prime?

Key findings

• For all $\varepsilon > 0$, the least prime represented by any given positive definite binary quadratic form of discriminant $q$ is bounded by $|q|^{7+\varepsilon}$ for all forms to 'most' discriminants.
• For 'most' forms to 'most' discriminants, the least prime represented is bounded by $|q|^{3+\varepsilon}$, improving the exponent in the unconditional bound.
• The Bombieri–Vinogradov-type result (1.3) holds for $Q^{20/3 + \varepsilon} \leq X(\log X)^{-B}$, with the error term bounded by $Q^{1/2}X(\log X)^{-A}$.
• The Barban–Davenport–Halberstam-type result (1.4) holds for $Q^{3 + \varepsilon} \leq X(\log X)^{-2A - 4}$, with the mean-square error bounded by $Q^{1/2}X^2(\log X)^{-A}$.
• Under the Lindelöf Hypothesis, the exponent in the bound for the least prime can be reduced from $7+\varepsilon$ to $2+\varepsilon$ and from $3+\varepsilon$ to $1+\varepsilon$, respectively.
• Conditional on $y_{\min} = 1$ for almost all squarefree $n$, the least prime of the form $x^2 + ny^2$ satisfies $p_0(n) \leq n^{2+\varepsilon}$ for almost all $n$.