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arxiv: 2606.07480 · v1 · pith:5MOVS2J2new · submitted 2026-06-05 · 🧮 math.NT · math.PR

ErdH{o}s-Kac theorems for discriminants of number fields

Jack B. Miller This is my paper

Pith reviewed 2026-06-27 20:44 UTC · model grok-4.3

classification 🧮 math.NT math.PR
keywords Erdős-Kac theoremramified primesabelian Galois extensionsnumber fieldsdiscriminantscentral limit theoremTauberian theoremsEuler products
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The pith

An Erdős-Kac theorem holds for the number of ramified primes in random abelian Galois extensions of number fields, with dependent events at distinct primes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the number of distinct ramified primes in a random abelian G-extension follows a normal distribution as the discriminant grows, mirroring the classical Erdős-Kac result on prime factors of integers. This supplies the first explicit examples in which the local ramification indicators at different primes are statistically dependent rather than independent. The argument rests on new general probability theorems that deliver central limit theorems for additive functions on sequences of ideals, provided those sequences obey Tauberian conditions expressible as finite sums of Euler products. These tools recover the earlier S_d results for small d as special cases and apply directly to the discriminant ideal in abelian extensions.

Core claim

We prove an analog of the Erdős-Kac theorem for the number of ramified primes in a random G-extension of a number field when G is abelian. This 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 Erdős-Kac theorems for sequences of ideals in a number field, subject to Tauberian hypotheses involving finite sums of Euler products.

What carries the argument

General central-limit theorems for additive arithmetic functions on sequences of ideals whose Dirichlet series are finite sums of Euler products, applied to the ramification-counting function on abelian extensions.

If this is right

  • The count of ramified primes in abelian G-extensions is asymptotically normally distributed with mean and variance both growing like log log of the discriminant.
  • The indicator functions for ramification at two distinct primes have nonzero limiting covariance when G is abelian.
  • The same probability machinery yields Erdős-Kac theorems for any other sequence of ideals whose zeta functions are finite sums of Euler products.
  • The dependence phenomenon is tied to the abelian case and does not appear in the previously studied S_d extensions for d ≤ 5.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dependence may be traceable to the explicit description of abelian extensions via class-field theory, which correlates ramification across primes through the conductor.
  • One could test whether the Tauberian conditions hold for other natural sequences, such as conductors of elliptic curves or discriminants of non-Galois extensions.
  • The result suggests that the independence seen in non-abelian cases is not automatic and may fail for other families once suitable Tauberian data become available.

Load-bearing premise

The sequences of ideals must satisfy Tauberian hypotheses involving finite sums of Euler products for the out-of-the-box probability results to apply.

What would settle it

An explicit computation, for a fixed small abelian group G and a sequence of number fields with growing discriminant, showing that the normalized count of ramified primes fails to converge in distribution to a standard normal random variable.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves an analogue of the Erdős-Kac theorem for the number of ramified primes in a random abelian G-extension of a number field. It develops reusable probability results for sequences of ideals satisfying Tauberian hypotheses phrased in terms of finite sums of Euler products, and applies these to obtain a central limit theorem while showing that local ramification events at distinct primes are dependent; this extends the Lemke Oliver–Thorne results for G = S_d (2 ≤ d ≤ 5).

Significance. If the Tauberian hypotheses hold for the relevant ideal sequences, the work supplies the first explicit examples of dependence among local ramification conditions, which bears on the distribution of discriminants. The out-of-the-box probability machinery for ideal sequences is a reusable contribution that strengthens the analytic toolkit in this area.

major comments (2)
  1. [Main theorem and application to number fields] The central application (proof of the analogue for abelian G-extensions): the manuscript must explicitly verify that the sequences of ramified ideals arising from abelian G-extensions satisfy the stated Tauberian hypotheses involving finite sums of Euler products. This verification is load-bearing for both the CLT and the claimed non-independence of local events; without it the general theorems do not apply.
  2. [Probability results for ideal sequences] The probability machinery section: the derivation of the CLT and dependence from the Euler-product form of the hypotheses is presented as out-of-the-box, but the error terms and the precise matching of the abelian-extension sequences to the required finite-sum shape need to be checked in detail, as any mismatch would invalidate the dependence statement.
minor comments (2)
  1. Notation for the Tauberian conditions and the associated Euler products could be made more self-contained, with a short reminder of the hypotheses before each application.
  2. A brief comparison table or paragraph contrasting the independence in the S_d cases with the dependence obtained here would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The points raised concern the explicitness of the verification that abelian G-extension ramification sequences satisfy the Tauberian hypotheses and the details of matching/error terms in the probability machinery. We agree these aspects benefit from greater clarity and will revise the manuscript to address them directly while preserving the core results.

read point-by-point responses

  1. Referee: [Main theorem and application to number fields] The central application (proof of the analogue for abelian G-extensions): the manuscript must explicitly verify that the sequences of ramified ideals arising from abelian G-extensions satisfy the stated Tauberian hypotheses involving finite sums of Euler products. This verification is load-bearing for both the CLT and the claimed non-independence of local events; without it the general theorems do not apply.

    Authors: We agree that explicit verification is essential. The manuscript already derives the relevant Euler products from the Artin conductors and class field theory for abelian G (see Section 4), showing they take the required finite-sum form over characters of G. To strengthen this, we will add a dedicated subsection that states the hypotheses verbatim, confirms each condition holds with explicit constants, and cross-references the resulting CLT and dependence statements. This is a clarification rather than a change in the argument. revision: yes

  2. Referee: [Probability results for ideal sequences] The probability machinery section: the derivation of the CLT and dependence from the Euler-product form of the hypotheses is presented as out-of-the-box, but the error terms and the precise matching of the abelian-extension sequences to the required finite-sum shape need to be checked in detail, as any mismatch would invalidate the dependence statement.

    Authors: The general CLT and dependence theorems are proved from the finite-sum Euler-product hypotheses with error terms controlled by the Tauberian assumptions (Proposition 2.3 and Theorem 2.5). For the abelian case the matching is exact because the local conditions factor through the finite group G and yield a finite linear combination of Dirichlet L-functions. We will insert a detailed verification paragraph (with explicit error bounds) immediately after the statement of the general theorems to confirm the shape and error control, thereby securing the dependence claim. revision: yes

Circularity Check

0 steps flagged

No circularity: new probability results developed independently and applied under explicit Tauberian hypotheses

full rationale

The paper develops general probability machinery for sequences of ideals satisfying Tauberian conditions phrased as finite sums of Euler products, then applies it to ramification in abelian G-extensions. This is not a self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain; the central Erdős-Kac analog and non-independence claim follow from the form of the Euler products once the hypotheses are granted. No quoted step reduces the main theorem to its inputs by construction. The work cites Lemke Oliver–Thorne for the non-abelian cases but introduces new results for the abelian setting.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on Tauberian hypotheses involving finite sums of Euler products for sequences of ideals; these are domain assumptions required for the probability machinery to apply.

axioms (1)
  • domain assumption Sequences of ideals satisfy Tauberian hypotheses involving finite sums of Euler products.
    The probability results are stated to apply subject to these hypotheses.

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