The typical elasticity of a quadratic order

TL;DR

This paper investigates the elasticity of orders in quadratic number fields, revealing asymptotic formulas for almost all split-free conductors, and introduces new statistical methods and inequalities in number theory.

Contribution

It provides new asymptotic estimates for the elasticity of quadratic orders and develops a weighted Turán--Kubilius inequality for analyzing split-free integers.

Findings

For imaginary quadratic fields, elasticity grows roughly as f divided by a power of log f.

For real quadratic fields, elasticity behaves like a power of log f under GRH.

New statistical theorems about class groups and a weighted Turán--Kubilius inequality are introduced.

Abstract

For an atomic domain $D$, the $elasticity$ $\rho(D)$ of $D$ is defined as $\sup\{r/s: \pi_1\cdots \pi_r = \rho_1 \cdots \rho_s,~ \text{where each $\pi_i, \rho_j$ is irreducible}\}$; the elasticity provides a concrete measure of the failure of unique factorization in $D$. Fix a quadratic number field $K$ with discriminant $\Delta_K$, and for each positive integer $f$, let $\mathcal{O}_f = \mathbb{Z} + f\mathcal{O}_K$ denote the order of conductor $f$ in $K$. Results of Halter-Koch imply that $\mathcal{O}_f$ has finite elasticity precisely when $f$ is $\textit{split-free}$, meaning not divisible by any rational prime $p$ with $(\Delta_K/p)=1$. When $K$ is imaginary, we show that for almost all split-free $f$, \[ \rho(\mathcal{O}_f) = f/(\log{f})^{\frac{1}{2}\log\log\log{f} + \frac{1}{2}C_K+o(1)}, \] for a constant $C_K$ depending on $K$. When $K$ is real, we prove under the assumption of the Generalized Riemann Hypothesis that \[ \rho(\mathcal{O}_f)= (\log{f})^{\frac12 +o(1)} \] for almost all split-free $f$. Underlying these estimates are new statistical theorems about class groups of orders in quadratic fields, whose proofs borrow ideas from investigations of Erd\H{o}s, Hooley, Li, Pomerance, Schmutz, and others into the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^\times$. One novelty of the argument is the development of a weighted version of the Tur\'{a}n--Kubilius inequality to handle a variety of sums over split-free integers.