The typical elasticity of a quadratic order

Abstract

For an atomic domain $D$, the $elasticity$ $\ddot{I}(D)$ of $D$ is defined as $\sup\{r/s: π_1\cdots π_r = ρ_1 \cdots ρ_s,~ \text{where each $π_i, ρ_j$ is irreducible}\}$; the elasticity provides a concrete measure of the failure of unique factorization in $D$. Fix a quadratic number field $K$ with discriminant $\hat{I}{”}_K$, 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 $\text{split-free}$, meaning not divisible by any rational prime $p$ with $(\hat{I}{”}_K/p)=1$. When $K$ is imaginary, we show that for almost all split-free $f$, \[ ρ(\mathcal{O}_f) = f/(\log{f})^{\frac{1}{2}\log\log\log{f} + \frac{1}{2}C_K+o(1)}, \] for a constant $C_K$ depending on $K$. When $K$ is real, we prove under the assumption of the Generalized Riemann Hypothesis that \[ ρ(\mathcal{O}_f)= (\log{f})^{\frac12 +o(1)} \] 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 Erdős, 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 Turán--Kubilius inequality to handle a variety of sums over split-free integers.

55 pages