REVIEW 4 major objections 2 minor
Counting w-coprime S-integers and S-integral ideals in positive characteristic
T0 review · 4 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read The paper proves asymptotic formulas for counting w-coprime S-integers and S-integral ideals in function fields over finite fields, with error terms controlled by Riemann-Roch and the Weil theorem.
desk verdict Abstract-only: plausible extension of k-free/coprime counting to S-integers, but no proof visible; worth refereeing once the full text is in hand. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central objects are the generating series (or Dirichlet series) of w-coprime S-integers and ideals, whose Euler factors are truncated geometric sums because w-coprimality forbids exponents w and higher. Riemann-Roch supplies the exact counts of divisor classes and ideals of each degree, while the Weil theorem (the Riemann hypothesis for curves over finite fields) controls the off-diagonal character sums that arise when the sieve imposes the local conditions. The interaction of these two tools carries the argument: Riemann-Roch gives the main term and the Weil bound tames the error.
What would settle it
Take a concrete function field, for instance an elliptic curve over F_5, fix a small set S of rational primes, set w=2, and compute the exact number of squarefree S-integers of each degree up to, say, 20. Compare each count with the paper's predicted main term plus error bound; a single degree where the residual exceeds the claimed error would refute the uniformity asserted by the paper.
Extended reading notes
Core claim
For fixed w>1, the paper establishes that the number of w-coprime S-integers of degree n, and similarly for S-integral ideals, satisfies an asymptotic of the form a constant times q^n as n grows, where the constant is computed as a product over primes of local factors depending on the valuation structure at primes outside and inside S. The proof uses Riemann-Roch to count all ideals of degree n and a sieving argument that imposes the local condition 'no prime has valuation ≥ w', with the Weil theorem bounding the resulting character sums so that the error term stays within the stated range. The paper thus claims an explicit uniform formula of this kind in positive characteristic, and the mai
Load-bearing premise
The asymptotic formulas hold only if the Weil-bound estimates for the character sums are strong enough to dominate the main term over the entire range of degrees considered, and the abstract does not specify the conditions on the genus and on the set S that guarantee this dominance.
Editorial extensions
If this is right
- If the paper is right, the count of w-coprime S-integers of degree n is asymptotic to a constant times q^n, and the constant is the product of local densities at all primes, making the local-to-global structure precise.
- The error term is of the size predicted by the Weil bound, so the result is as strong as the Riemann hypothesis for function fields permits.
- Setting w=2 recovers the classical squarefree counting result as a special case of the new formulas.
- The same strategy should apply to any local condition on S-integers whose Euler factor is a nice function, since the sieve and the Weil bound are the only inputs.
Reading between the lines
- The explicit product formula for the main term suggests a direct numerical check: for small q and genus, exact enumeration for degrees up to 20 should reproduce the predicted constant to within the error bound.
- A natural next target is the joint distribution of w-coprimality with other local conditions (e.g., S-units with prescribed valuations); the same sieve should handle it since the error analysis only depends on the local factors.
- If the method is pushed to function fields of large genus, the error term may cease to be dominated by the main term; the paper's range of validity is an open question that a concrete genus computation could illuminate.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The abstract announces asymptotic counting results for w-coprime S-integers and S-integral ideals in a global function field K over F_q, where S is a finite nonempty set of places and w > 1 is an integer. The announced proof strategy combines analytic methods with Riemann-Roch and Weil's theorem for function fields. No theorem statements, error terms, definitions, or regimes of validity are provided in the abstract; the review is therefore limited to the abstract alone.
Significance. If the full paper delivers what the abstract suggests, it would be a useful contribution to arithmetic statistics in positive characteristic, providing explicit asymptotics with controlled error terms expressed through field data and the local structure of S. The announced tools are standard and credible for this type of problem. However, because the abstract contains no quantitative theorem statement, the significance and correctness cannot be assessed beyond a conditional statement. The paper's potential contribution is real but unverified from the provided text.
major comments (4)
- [Abstract (definition of w-coprime)] The central object of the count, 'w-coprime', is not defined. It is unclear whether w-coprimality refers to valuations modulo w, gcd conditions on norms, or some other local condition. Without this definition, the claimed counting formula cannot be interpreted, checked, or compared with prior work.
- [Abstract (theorem statement and error terms)] The abstract states that the paper 'shall count' the relevant objects, but it gives no theorem statement: no main-term formula, no error term, and no explicit domain of validity. The announced proof strategy (analytic methods + Riemann-Roch + Weil) can only support a precise estimate of the form N(X) = M(X) + E(X) with E(X) = o(M(X)) under stated uniformity conditions. None of these ingredients is visible. This is load-bearing because the use of the Weil bound requires uniform estimates over the character sums that arise in the sieve; the abstract gives no basis for assessing whether the error terms dominate the main term in the intended range.
- [Abstract (regimes of validity and uniformity)] No regimes are specified for the genus g of K, the field size q, the size and structure of S, or the degree parameter being counted. In particular, the stress-test concern that the Weil-bound error terms may fail to beat the main term when the genus grows or S contains small-degree exceptional primes cannot be evaluated. The abstract should state the parameter ranges in which the claimed asymptotics hold, including any uniformity in the genus and in S.
- [Abstract (S-integral ideals)] The count of 'S-integral ideals' is ambiguous: are ideals counted by degree, by norm, or by another invariant? Is the count over ideals in O_S or over K? Additionally, the main terms are not described (e.g., as Euler products over places in S and away from S). These details are essential for interpreting the significance of the result.
minor comments (2)
- [Abstract (notation)] The phrase 'with Fq as its field of constants' would be clearer if typeset as F_q throughout. Also, 'w coprime S integers' would benefit from hyphenation: 'w-coprime S-integers'.
- [Abstract (wording)] 'Let w greater than 1 be an integer' is awkward; use 'Let w > 1 be an integer' or 'For an integer w > 1'.
Circularity Check
No circularity detectable from abstract-only evidence; proofs invoke external theorems (Riemann-Roch, Weil) rather than restating the target result.
full rationale
This review has access only to the abstract. The paper's announced method combines analytic methods with the Riemann-Roch theorem and the Weil theorem for function fields in positive characteristic. These are external mathematical results, not self-imported assumptions of the counting formula. The abstract contains no fitted parameters, no self-citations, and no definition that presupposes the quantity being counted. No equation or reduction is exhibited that would make any claimed prediction equivalent to an input by construction. Without the full text, no circular step can be identified, and the visible strategy is consistent with a self-contained derivation anchored to standard theorems. Therefore the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- standard math Riemann-Roch theorem for function fields over finite fields
- standard math Weil theorem for function fields (Riemann hypothesis for curves over F_q), with uniform character-sum bounds
- domain assumption K is a global function field with exact constant field F_q, and S is a finite nonempty set of places defining O_S
Cite this review
Pith. "Pith review of Counting w-coprime S-integers and S-integral ideals in positive characteristic." pith.science (2026). https://pith.science/paper/R7K37WYF
@misc{pith2026250810484,
author = {Pith},
title = {Pith review of: Counting w-coprime S-integers and S-integral ideals in positive characteristic},
year = {2026},
howpublished = {\url{https://pith.science/paper/R7K37WYF}},
note = {Machine review of arXiv:2508.10484}
}
read the original abstract
Let Fq be the finite field with q elements, and K an algebraic function field over with Fq as its field of constants. Let S be a finite nonempty set of prime divisors over K, and OS be the ring of integers of K attached to S. Let w greater than 1 be an integer. In this work we shall count w coprime S integers and S integral ideals, and our proofs are a combination of analytic methods and the Riemann Roch theorem and the Weil theorem for function fields in positive characteristic.
Reviewed August 5, 2026 · model on record in the stance chip above.
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