On the role of higher roots in prime ideal races

Mounir Hayani

REVIEW 3 major objections 3 minor 16 references

On the role of higher roots in prime ideal races

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that prime ideal races can be biased by a difference in higher-order root counts alone, and pins the smallest Galois group orders at 96 and 320.

desk verdict New higher-root mechanism for prime ideal bias is real and worth engaging, but the sharp n_5=320 minimality claim currently rests on an appendix lemma that is not proved as written. read the letter →

arxiv 2607.23150 v1 pith:2OIA3XDC submitted 2026-07-25 math.NT

Mounir Hayani This is my paper

classification math.NT MSC 11R4211N1311R4520D2020C15

keywords Chebyshev'sbiasprimeidealracesArtinL-functionsroot-countingfunctiongeneralizedquaterniongroupsminimalGaloisgrouporderlogarithmicdensityWall

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

For a Galois extension L/K, the paper asks when the prime ideal race between two conjugacy classes is biased, meaning the logarithmic density of the set where one class leads exceeds 1/2. Earlier examples tied such bias to square-root counts or to zeros of Artin L-functions at s=1/2. The paper introduces algebraic parameters that isolate the first root level where the two classes differ, proves a conditional criterion for bias in terms of those parameters, and then constructs extensions where the only difference is in the number of 2p-th roots, with no contribution from square roots or vanishing L-functions. For p=3 and p=5 it shows the Galois group order is minimal: 96 and 320, realized by products of generalized quaternion groups. It also extracts an unconditional proof, in certain cases, of an asymptotic previously proved under the Deep Riemann Hypothesis.

What carries the argument

The engine is the pair of algebraic parameters ι_N(g) and γ(g): for a class function g, ι is the smallest squarefree ℓ for which the induced function Ind(g∘f_ℓ) is nonzero, and γ is the smallest squarefree ℓ for which ⟨g,r_ℓ⟩≠0. The bias criterion is carried by these parameters; the constructed examples depend on the property P(2p;x,y), which requires equal root counts for every squarefree ℓ<2p but strictly fewer 2p-th roots of one class than the other. The group-theoretic constructions use generalized quaternion groups Q_{4n} embedded in GL(4,F_q) and then in S_{q^4}; the lower-bound proofs use the Schur–Zassenhaus theorem, root-counting formulas such as r_{2p}(g)=p·r_2^H(g), and Wall's cla

What would settle it

A computer search over all finite groups of order <96 can settle the p=3 claim: if any group has elements x,y of the same order with equal root counts for every squarefree ℓ<6 (in effect r_2 and r_3 equal) but r_6(x)<r_6(y), then Theorem 4.2's bound is false; similarly for p=5 with ℓ<10 and order <320. On the analytic side, computing δ for the constructed Q_12×Q_8 extension under the stated hypotheses and finding δ=1/2 would falsify Corollary 1.3's application.

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Extended reading notes

Core claim

The paper's central claim is that Chebyshev-type bias in prime ideal races can be generated entirely by the counts of 2p-th roots, for an odd prime p, while the competing classes have identical square-root counts and all Artin L-functions of the Galois closure over Q are nonvanishing at s=1/2. Theorem 1.2 gives a conditional characterization of when δ_{L/K}(t) lies in (1/2,1) in terms of the parameters ι(t) and γ(t); Corollary 1.3 specializes this to the case where bias is equivalent to γ(t_{C1,C2})=2ι(t_{C1,C2}) and μ(2ι)(r_{2ι}(C1)-r_{2ι}(C2))<0. Theorem 1.5 realizes this for every odd prime p: extensions L/K exist with Gal(L/Q) ≅ S_{q^4}, q≡1 mod 2p(p−1), and Gal(L/K) ≅ Q_{4p}×Q_{4(p−1)}×

Load-bearing premise

The sharp minimal-order claims for p=3 and p=5 rest on the technical lower-bound argument in Section 4, especially the use of Wall's classification of groups with more than half involutions and the separate elimination of groups of orders 240 and 270 for p=5; if that classification misses a case, the bounds n_3=96 and n_5=320 could fail even though the constructions remain valid.

Editorial extensions

If this is right

If the paper's criterion is correct, the standard picture of what drives prime ideal races is incomplete: higher-degree root counts can be the sole cause of bias, independent of square roots and zeros at s=1/2.
The minimal orders n_3=96 and n_5=320 become concrete benchmarks: any extension with Galois group below these orders cannot exhibit this purely higher-root bias for p=3 or p=5.
For every odd prime p there exist extensions with bias in (1/2,1) whose Galois group over Q is a symmetric group S_{q^4}, which has no irreducible symplectic representation, so the nonvanishing condition at s=1/2 is satisfied unconditionally in the representation-theoretic sense.
The same parameters give unconditional cases of the Aoki–Koyama log-log asymptotic, and for p≥2 produce extreme biases where the unweighted difference has a constant sign for all sufficiently large x.
The criterion also characterizes when the signed difference between the two prime counting functions changes sign infinitely often: it does so exactly when the logarithmic density lies in (0,1).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the same root-counting mechanism likely works for biases driven by d-th roots for any squarefree d with at least two prime factors, not only d=2p; the parameters ι and γ point to a general setting where the bias is located at the first squarefree level at which induced root counts differ.
Beyond the paper: the conjectured formula n_p = min_{n∈N_p} 16n(n−1)P_n, stated in the paper as an expectation, might be approachable layer by layer using the lower-bound machinery; verifying it for p=7 would test the Wall-type classification step in a new case.
Beyond the paper: the unconditional Aoki–Koyama cases suggest that the Deep Riemann Hypothesis is not a genuine prerequisite for log-log-size biases in these races; one might expect the conditional hypotheses in the main criterion to be similarly removable for a wider class of extensions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new mechanism is genuine. The paper's core contribution—bias driven purely by 2p-th root counts, with equal square-root counts and no s=1/2 vanishing—is absent from the prior literature, and the conditional characterization via iota and gamma in Theorem 1.2 is a useful unifying framework. The constructions for all odd primes p, using Q_{4p} × Q_{4(p−1)} × C_{rad(p−1)/2} with an S_{q^4} embedding, are clever and the arithmetic side (no symplectic representations, inverse Galois via S_{q^4}) checks out. Nothing here is fitted; iota and gamma are exact algebraic quantities.

The soft spots are concentrated in the lower-bound machinery. The reader's conditional verdict is fair, but I would go further: the stress-test note identifies a specific, load-bearing flaw in Lemma 4.25 in the appendix. The claim that a group of order 12 cannot contain elements of order 3 because the listed generators have even order is simply wrong. The later assertion 'By Lemma 4.7, this forces Ord(y)=2' also does not follow when Ord(y)=4, since 16 divides 240. These exclusions are necessary for Theorem 4.2 and hence for n_5=320. Maybe a Sylow/cardinality argument repairs the lemma, but that argument is not in the paper. The general bound |G| ≥ 8p^2 in Theorem 4.1 looks credible from the parts I checked, but the sharp p=5 bound is not confirmed as written.

The analytic sections are also abbreviated—'we keep the proofs short' in Section 2.2, and the embedding proposition is a five-step outline. This raises the verification burden but is not a flaw in itself, since the references to [8,13] are standard.

Who gets value: anyone working on Chebyshev's bias or prime ideal races. The paper deserves a serious referee—the main ideas are important and likely correct—but the referee should require either a complete proof of Lemma 4.25 or a softened statement of the n_5=320 claim. I would cite the constructions and the criterion; I would not cite the minimality result until the appendix is repaired.

Referee Report

3 major / 3 minor

Summary. The paper introduces two algebraic parameters, ι(t) and γ(t), attached to class functions on the Galois group of a prime ideal race, and proves a conditional characterization (Theorem 1.2) of when the Rubinstein–Sarnak logarithmic density δ_{L/K}(C1,C2) lies in (1/2,1). The main arithmetic novelty is the construction, for every odd prime p, of Galois extensions L/K such that the bias is produced neither by square-root counts nor by zeros of Artin L-functions at s=1/2, but entirely by a difference in the number of 2p-th roots. The paper also proves lower bounds on the Galois group order, in particular n_3=96 and n_5=320 for the minimal extensions, using a group-theoretic analysis of groups satisfying the property P(2p;x,y). Finally, Theorem 1.6 and Theorem 1.7 give unconditional asymptotic formulae for weighted prime ideal races, refining and extending work of Aoki–Koyama and of the author's earlier paper.

Significance. If the main results are correct, the paper gives the first systematic mechanism for Chebyshev bias that is driven solely by higher-root counts, and it supplies explicit, parameter-free algebraic criteria. The minimality statements n_3=96 and n_5=320 are attractive and appear to be supported by the bulk of the §4 analysis. The construction via generalized quaternion groups and their embeddings into GL(4,F_q) is explicit and checkable, and the analytic framework in §2 follows the well-established Rubinstein–Sarnak/Fiorilli–Jouve machinery. The unconditional extreme-bias results in Theorem 1.6/1.7 are also a genuine strengthening. The paper is therefore potentially a solid contribution to the subject, provided the load-bearing lower-bound argument for p=5 is repaired.

major comments (3)

[Appendix, Lemma 5.1] The step after showing that the subgroup generated by the square roots of y has order at least 12 is not valid as written. The text says it cannot have order 12 'because among the previously listed elements, there is none of order 3.' A group of order 12 can be generated by elements of even order and still contain elements of order 3 (for example A4). This gap is load-bearing: Lemma 5.1 is used to exclude |G|=240 in the proof of Lemma 4.25, and Lemma 4.25 is essential for the claimed minimality n_5=320 in Theorem 4.2/Theorem 1.5. A correct Sylow/cardinality argument, or a direct machine verification for the groups of order 240, is needed.
[Appendix, proof of Lemma 4.25] After proving |N_G(P)|=40, the proof states: 'By Lemma 4.7, this forces Ord(y)=2.' Lemma 4.7 only implies that 16 divides |G| when Ord(y) is a power of 2. Since 16 divides 240, the case Ord(y)=4 is not excluded by Lemma 4.7. The subsequent 2-Sylow analysis therefore does not rule out Ord(y)=4, and the exclusion of |G|=240 is incomplete. This is another load-bearing point for the sharp bound n_5=320.
[Appendix, Lemma 4.25, step (4)] The proof also asserts that s has order 2Ord(y) ∈ {4,8} and then immediately concludes Ord(y)=2. The argument only excludes Ord(y)=1 (impossible) and Ord(y)=? if Ord(s)=8 then Ord(y)=4, and no contradiction is supplied. This is a concrete manifestation of the same gap as the previous comment: the possible order-4 case for y is not handled.

minor comments (3)

[§4.4, proof of Corollary 4.24] The count 'at least ⌊5p/4⌋+1 square roots of 1' in N/<y> appears to be a typo. The preceding argument gives at least p + (⌊p/2⌋+1) = ⌊3p/2⌋+1 involutions. The weaker bound still exceeds 5/8 of the order, so the conclusion is unaffected, but the displayed count should be corrected.
[§3.3, Proposition 3.11] In Case 1, the displayed inequality '4(n−1)+4p_j ≥ r_{2p_j}(x)' is an equality. The surrounding comparison would be clearer if written as '4np_j > 4(n−1)+4p_j = r_{2p_j}(x)'.
[§1.1, after Theorem 1.5] The statement that the constructed group has order 16p(p−1) is only true when rad(p−1)=2 (in particular for p=3,5). For general p the order is 8p(p−1)rad(p−1). The text should make this qualification explicit to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation chain is self-contained.

full rationale

The paper's central chain is not circular. Definition 1.1 defines the algebraic parameters ι and γ directly from induced class functions and root counts, not from the limiting density δ; Theorem 1.2 derives the criterion for δ∈(1/2,1) via explicit formulas and the standard Rubinstein–Sarnak limiting-distribution machinery, with no fitted parameters. The group-theoretic construction of E_p extensions is unconditional: Proposition 3.11 computes root counts in Q_{4n}×Q_{4(n−1)}×C_{p_1...p_{j−1}} directly, and Proposition 3.17 proves the required property Q(2p;x,y) by an eigenvalue/cycle-type analysis, with the biased density then following from Corollary 1.3. E_d is indeed defined so that Corollary 1.3 applies, but non-emptiness of E_p is a separate theorem, not assumed by the definition. The minimality claims n_3=96 and n_5=320 are lower-bound results proved from Wall's external classification [16] and the lemmas in §4; they do not assume the constructed groups. Self-citations to [9] occur for auxiliary results (Proposition 3.3, parts of Theorem 1.7) and are explicit, published, and independent of the main E_p/minimality claims; they are not load-bearing for the central derivation. The appendix proof of Lemma 4.25 contains at least one questionable justification (‘because among the previously listed elements, there is none of order 3’), but a logical gap in an exclusion argument is a correctness risk, not a circular reduction to the paper's own inputs. It therefore does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No fitted constants appear anywhere: p, q, n are construction variables chosen by existence theorems (Dirichlet primes q ≡ 1 mod 2p(p−1)), not tuned to data. The new objects — iota, gamma, properties P(2d;x,y) and Q(2d;x,y) — are exact algebraic definitions with computable content, not postulated entities. All bias-conclusion axioms are the field's standard GRH/AC/LI hypotheses; the lower-bound proofs load two external pillars (Wall's classification, Schur–Zassenhaus).

assumptions (8)

domain assumption GRH(L): all non-trivial zeros of the Artin L-functions of L/Q lie on Re(s)=1/2
Invoked in Theorem 1.2, Corollaries 1.3–1.4, Propositions 1.8, 2.2, 2.4; every delta in (1/2,1) conclusion is conditional on it.
domain assumption AC(L): Artin's holomorphicity conjecture, L(s;L/Q;chi) entire for all non-trivial irreducible chi
Invoked in Theorem 1.2 and Proposition 2.4 through the explicit formula [8, Lemma 3.16].
domain assumption LI−(L) and LI(L): linear independence of positive zero imaginary parts, plus non-vanishing at s=1/2 for orthogonal and unitary characters
Used in Theorem 1.2 and Corollary 1.4; the non-vanishing half is conjectural.
domain assumption No irreducible symplectic representation of Gal(L/Q) implies no Artin L-function vanishes at s=1/2
The definition of E_d and Corollary 1.4 rely on this (cited [8, Section 1.2, Conjecture (LI)]); the constructed G+ = S_{q^4} has no non-trivial symplectic irreducibles.
standard math Wall's classification (1970): finite groups with r_2(1) > |G|/2 are E × G_0 with G_0 of types I–IV
Theorem 4.14 (cited [16]); the lower-bound argument (Lemmas 4.15, 4.16, Proposition 4.22, Corollary 4.24) depends on it.
standard math Schur–Zassenhaus theorem on complements of normal Sylow subgroups
Theorem 4.10 and Corollary 4.11; used throughout Section 4 to move x, y into a fixed complement H.
standard math Lagarias–Odlyzko effective Chebotarev theorems (conditional and unconditional forms)
Supplies the bounds psi(x) << x^{1/2} and psi(x) = x + O(...) used in Propositions 2.2 and Lemma 2.3.
standard math In the Cayley embedding of G into S_n, elements of the same order are conjugate
Invoked in Proposition 3.2 to obtain gamma = iota for Theorem 1.7; the proof is deferred to [9, Proof of Theorem 1.2, Section 2].

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Cite this review

Pith. "Pith review of On the role of higher roots in prime ideal races." pith.science (2026). https://pith.science/paper/2OIA3XDC

@misc{pith2026260723150,
  author       = {Pith},
  title        = {Pith review of: On the role of higher roots in prime ideal races},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OIA3XDC}},
  note         = {Machine review of arXiv:2607.23150}
}

abstract

Let $L/K$ be a Galois extension of number fields, and let $C_1, C_2$ be conjugacy classes of $\mathrm{Gal}(L/K)$. By introducing algebraic parameters related to $C_1$ and $C_2$, we provide a conditional characterization for Chebyshev's bias logarithmic density $\delta_{L/K}(C_1,C_2)$ to lie in the interval $(1/2, 1)$, meaning that the prime ideal race is biased. Unlike existing examples in the literature, where the bias comes from a difference in the number of square roots, the order of vanishing of Artin $L$-functions at $s=1/2$, or a combination of both, we use this criterion to construct Galois extensions where neither of these aspects plays a role. In our constructions, the bias arises entirely from a difference in the number of $2p$-th roots for an odd prime $p$. We prove that the minimal Galois group order required for this phenomenon is $96$ for $p=3$ and $320$ for $p=5$, where in both cases the group structure is a direct product of two generalized quaternion groups. Furthermore, we provide generalized constructions for all odd primes. Finally, these same algebraic parameters enable us to prove that an estimate established by Aoki and Koyama under the Deep Riemann Hypothesis holds unconditionally in certain cases.

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Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Aoki and S.-y

M. Aoki and S.-y. Koyama,Chebyshev’s bias against splitting and principal primes in global fields, J. Number Theory245(2023), 233–262, DOI 10.1016/j.jnt.2022.10.005. MR4517481

work page doi:10.1016/j.jnt.2022.10.005 2023

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Bailleul,Chebyshev’s bias in dihedral and generalized quaternion Galois groups, Algebra Number Theory15(2021), no

A. Bailleul,Chebyshev’s bias in dihedral and generalized quaternion Galois groups, Algebra Number Theory15(2021), no. 4, 999–1041

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A. Bailleul and M. Hayani,Prime ideal races with several competitors, arXiv preprint arXiv:2508.04087 (2025)

arXiv 2025

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P. Chebyshev,Lettre de M. le Professeur Tch´ ebychev ` a M. Fuss sur un nouveau th´ eor` eme relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3, Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg (1853), 208

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L. Devin,Chebyshev’s bias for analyticL-functions, Math. Proc. Cambridge Philos. Soc.169(2020), no. 1, 103–140

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Fiorilli and F

D. Fiorilli and F. Jouve,Unconditional Chebyshev biases in number fields, J. ´Ec. polytech. Math.9 (2022), 671–679

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,Distribution of Frobenius elements in families of Galois extensions, J. Inst. Math. Jussieu23 (2024), no. 3, 1169–1258

2024