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arxiv: 2603.20093 · v2 · submitted 2026-03-20 · 🧮 math.NT

A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races

Alexandre Bailleul , Mounir Hayani , Th\'eo Untrau This is my paper

Pith reviewed 2026-05-15 08:07 UTC · model grok-4.3

classification 🧮 math.NT
keywords Skewes numbersprime number racesWasserstein metricKronecker-Weyl theoremquadratic residuesgeneralized Riemann hypothesishighly composite moduli
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The pith

Sequences of highly composite moduli make generalized Skewes numbers grow rapidly enough to disprove Fiorilli's conjecture unconditionally.

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

The paper constructs sequences of highly composite moduli q for which the generalized Skewes numbers, marking the first sign change in the race between quadratic residues and nonresidues, grow very rapidly. This construction provides an unconditional disproof of a conjecture by Fiorilli. In addition, assuming the generalized Riemann hypothesis and an effective linear independence hypothesis on the zeros, the authors derive conditional upper bounds for these numbers. The approach relies on a quantitative version of the Kronecker-Weyl theorem expressed using the 1-Wasserstein metric to obtain explicit convergence rates to the limiting distributions.

Core claim

We study generalized Skewes numbers as the first locations where two comparable prime counting functions change sign. For the race between quadratic residues and quadratic nonresidues modulo q, we construct sequences of highly composite moduli q such that these Skewes numbers grow very rapidly, disproving unconditionally a conjecture of Fiorilli. Assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes numbers. Our method uses a quantitative Kronecker-Weyl theorem in the 1-Wasserstein metric to obtain explicit rates for convergence to the limiting distributions.

What carries the argument

A quantitative Kronecker-Weyl theorem formulated in terms of the 1-Wasserstein metric, used to derive explicit rates of convergence to limiting distributions in prime number races.

If this is right

  • The first sign changes in quadratic residue races can occur at arbitrarily large scales for certain sequences of q.
  • Fiorilli's conjecture on the bounded growth of generalized Skewes numbers is false.
  • Explicit convergence rates via the Wasserstein distance allow precise control over the distribution of prime counting discrepancies.
  • Conditional upper bounds on Skewes numbers hold when the generalized Riemann hypothesis and linear independence of zeros are assumed.

Where Pith is reading between the lines

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

  • The Wasserstein approach may extend to other prime number races involving different arithmetic progressions or characters.
  • Rapid growth for highly composite q suggests that prime distribution discrepancies can persist over longer intervals for specially chosen moduli.
  • The method could be tested numerically on small highly composite q to observe the predicted sign change locations directly.

Load-bearing premise

The existence of explicit sequences of highly composite moduli q making the first sign-change locations grow rapidly enough to contradict Fiorilli's conjecture.

What would settle it

A concrete computation of the first sign change for the smallest terms in one such sequence of q, showing whether the location exceeds the rapid growth required to contradict the conjecture.

read the original abstract

We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences of highly composite moduli $q$ such that those Skewes' numbers grow very rapidly in some sense. This disproves unconditionally a conjecture of Fiorilli. In the other direction, assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes' numbers. Our approach relies on a quantitative Kronecker-Weyl theorem formulated in terms of the $1$-Wasserstein metric to obtain explicit rates for the convergence to the limiting distributions in these races.

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 constructs sequences of highly composite moduli q such that the generalized Skewes numbers (first sign-change locations) in the quadratic residue/non-residue prime race grow rapidly enough to disprove Fiorilli's conjecture unconditionally. It also derives conditional upper bounds on these numbers assuming GRH together with an effective linear independence hypothesis on the zeros. The proofs rely on a quantitative Kronecker-Weyl theorem formulated in the 1-Wasserstein metric to obtain explicit convergence rates of the empirical measures on the torus to their limiting distributions.

Significance. If the explicit construction yields a concrete lower bound on the first crossing that exceeds the growth forbidden by Fiorilli's conjecture, the unconditional disproof would be a notable advance in the study of prime number races. The introduction of Wasserstein distances to control the speed of equidistribution offers a fresh quantitative tool that could extend to other Chebyshev-type biases. The conditional upper bounds supply effective rates under standard hypotheses and are of independent interest for explicit estimates in analytic number theory.

major comments (2)
  1. [§3] §3 (construction of the sequence q_n): the argument that the 1-Wasserstein convergence rate produces an explicit lower bound on the first sign change of the race function E(x;q,a) needs a precise translation step. The integrated deviation controlled by W_1 does not automatically guarantee that the deterministic orbit remains strictly positive (or negative) up to the claimed X; an additional uniform or tail estimate on the discrepancy appears necessary to convert the distributional rate into the required deterministic lower bound on the Skewes number.
  2. [§4.1] §4.1 (conditional upper bounds): the effective linear independence hypothesis is invoked to control the linear forms in the logarithms of the zeros, but the dependence of the resulting bound on the height of the zeros and on the compositeness of q should be made fully explicit so that the comparison with the unconditional lower bounds is quantitative.
minor comments (2)
  1. [Abstract] The abstract's phrase 'grow very rapidly in some sense' should be replaced by a concrete statement of the growth rate relative to Fiorilli's conjecture (e.g., log log X or exp(c sqrt(log X))).
  2. [§2] Notation for the residue/non-residue race function should be introduced once and used consistently; the transition from the torus measure to the prime-counting difference is not always clearly sign-posted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each of the major comments below and will revise the paper accordingly to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: §3 (construction of the sequence q_n): the argument that the 1-Wasserstein convergence rate produces an explicit lower bound on the first sign change of the race function E(x;q,a) needs a precise translation step. The integrated deviation controlled by W_1 does not automatically guarantee that the deterministic orbit remains strictly positive (or negative) up to the claimed X; an additional uniform or tail estimate on the discrepancy appears necessary to convert the distributional rate into the required deterministic lower bound on the Skewes number.

    Authors: We agree with the referee that a precise translation from the Wasserstein metric bound to a deterministic lower bound on the sign change is required. In the revised version, we will add a detailed explanation and an auxiliary estimate showing how the W_1 convergence, combined with the specific arithmetic properties of the chosen moduli sequence q_n, implies that the partial sums of the race function remain positive (or negative) up to the desired point. This will involve bounding the tail of the distribution using the effective equidistribution rate. revision: yes

  2. Referee: §4.1 (conditional upper bounds): the effective linear independence hypothesis is invoked to control the linear forms in the logarithms of the zeros, but the dependence of the resulting bound on the height of the zeros and on the compositeness of q should be made fully explicit so that the comparison with the unconditional lower bounds is quantitative.

    Authors: We thank the referee for this observation. We will revise §4.1 to state the effective linear independence hypothesis with explicit dependence on the height of the zeros and on the number of distinct prime factors of q. The resulting upper bounds will then be expressed with full explicit dependence on these quantities, enabling a direct and quantitative comparison with the unconditional lower bounds from the construction in §3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit construction and external theorem

full rationale

The paper's main unconditional result is an explicit construction of sequences of highly composite moduli q yielding rapidly growing generalized Skewes numbers, which directly contradicts Fiorilli's conjecture. This rests on a quantitative Kronecker-Weyl theorem in the 1-Wasserstein metric for convergence rates to limiting distributions on the torus; the theorem is classical and invoked as an external tool rather than derived internally or via self-citation. Conditional upper bounds invoke standard external hypotheses (GRH plus effective linear independence of zeros). No equation reduces a claimed prediction to a fitted input by construction, no ansatz is smuggled via self-citation, and no uniqueness theorem is imported from the authors' prior work. The derivation chain therefore contains independent mathematical content and does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The unconditional disproof relies on an explicit arithmetic construction of moduli; the conditional upper bounds rest on two standard but unproven hypotheses in analytic number theory.

axioms (2)
  • domain assumption Generalized Riemann Hypothesis
    Invoked to obtain the conditional upper bounds on generalized Skewes numbers.
  • domain assumption effective linear independence hypothesis on zeros of L-functions
    Required for the quantitative rates in the Wasserstein formulation of Kronecker-Weyl.

pith-pipeline@v0.9.0 · 5424 in / 1437 out tokens · 32005 ms · 2026-05-15T08:07:03.642638+00:00 · methodology

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