2024/11/13 by Paul Pollack, Fan, Steve, Pollack, Paul
Engineering · #Elasticity and Material Modeling
paper · pdf · doi:10.48550/arxiv.2411.09063
For an atomic domain D, the elasticity ρ(D) of D is defined as sup\r/s: π1⋯ πr = ρ1 ⋯ ρs,~ where each πi, ρj is irreducible\; the elasticity provides a concrete measure of the failure of unique factorization in D. Fix a quadratic number field K with discriminant ΔK, and for each positive integer f, let Of = ℤ + fOK denote the order of conductor f in K. Results of Halter-Koch imply that Of has finite elasticity precisely when f is split-free, meaning not divisible by any rational prime p with (ΔK/p)=1. When K is imaginary, we show that for almost all split-free f, ρ(Of) = f/(logf)^(1)/(2)logloglogf + (1)/(2)CK+o(1), for a constant CK depending on K. When K is real, we prove under the assumption of the Generalized Riemann Hypothesis that ρ(Of)= (logf)\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 (ℤ/mℤ)^×. 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.