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arXiv · 2606.07480

Erd\H{o}s-Kac theorems for discriminants of number fields

Jack B. Miller
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Abstract

The classical Erd\H{o}s-Kac theorem gives a central limit theorem for the number of prime divisors of a random integer. We prove an analog for the number of ramified primes in a random $G$-extension of a number field when $G$ is abelian. This builds on previous work of Lemke Oliver and Thorne in the cases $G = S_d$ ($2 \le d \le 5$), and provides the first examples where local ramification events at distinct primes are not independent. We develop probability results that can be used "out of the box" to prove Erd\H{o}s-Kac theorems for sequences of ideals in a number field, subject to Tauberian hypotheses involving finite sums of Euler products.

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Jack B. Miller. 2026-06-05. Erd\H{o}s-Kac theorems for discriminants of number fields. https://arxiv.org/abs/2606.07480

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