{"id":"65f6c7df-ff94-424d-a403-b806763fe81b","arxiv_id":"2508.10484","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper produces asymptotic counting formulas for w-coprime S-integers and S-integral ideals in function fields over finite fields, proved with Riemann-Roch and the Weil theorem.","lead":"This paper derives counting formulas for S-integers and S-integral ideals in function fields over finite fields that satisfy a 'w-coprime' condition, using analytic methods together with Riemann-Roch and Weil's theorem. A general reader might care because such exact counts transfer classical coprime-counting questions to positive characteristic, where the tools are cleaner.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No concrete flaw visible from abstract; the load-bearing condition is unverified uniformity of Weil/Riemann-Roch error terms in the stated asymptotics.","rationale":"The reader's weakest assumption—that the Weil bound and Riemann-Roch estimates must be uniform enough to make the error term smaller than the main term—is exactly the most load-bearing point. The abstract provides no evidence for this uniformity, and the definition of w-coprime is missing, so the claim cannot be verified from the abstract alone. However, the announced strategy is standard and credible: in positive-characteristic function fields, Weil's theorem gives sqrt(q)-saving bounds for character sums, and Riemann-Roch gives exact dimensions for divisor spaces, which typically yield asymptotic formulas of this type. Thus there is no demonstrated contradiction or obvious internal flaw. The correct disposition remains UNVERDICTED/UNCHANGED: the paper cannot be accepted or rejected without the full proof, and no concrete defect has been identified. The recommended test is a small computational cross-check once the full text is available, which would either confirm the error-term behavior or reveal a regime where the uniformity assumption breaks down.","tokens_in":810,"tokens_out":3073,"duration_ms":41460,"concrete_test":"Obtain the full text and isolate the theorem stating the asymptotic formula. For a concrete low-genus example (e.g., an elliptic or genus-2 function field over F_2 or F_3 with S containing one degree-1 prime and w = 2), substitute the claimed error term and compare with brute-force enumeration of w-coprime S-integers and S-integral ideals of degree d = 1,...,12. If the discrepancy exceeds the claimed error at any d in this range, the uniformity condition fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an asymptotic counting formula whose error term must be o(main term). The announced proof strategy (analytic methods + Riemann-Roch + Weil) could deliver this, but only if the character sums and Riemann-Roch counts are bounded uniformly in the parameters that appear in the theorem. The abstract does not specify the regimes (genus g, field size q, structure of S), nor does it define \"w-coprime\" or the associated local factors. If, for example, S contains degree-1 primes whose local factors coincide with the main-term growth, or if the genus grows with the degree parameter, the error term could fail to be dominated by the main term. This is not an internal inconsistency visible from the abstract; it is an unverified uniformity condition. Without the full theorem statements and proofs, no stronger conclusion can be drawn.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":868,"tokens_out":2578,"duration_ms":30634,"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":[{"comment":"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.","section":"Abstract (definition of w-coprime)"},{"comment":"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.","section":"Abstract (theorem statement and error terms)"},{"comment":"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.","section":"Abstract (regimes of validity and uniformity)"},{"comment":"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.","section":"Abstract (S-integral ideals)"}],"minor_comments":[{"comment":"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'.","section":"Abstract (notation)"},{"comment":"'Let w greater than 1 be an integer' is awkward; use 'Let w > 1 be an integer' or 'For an integer w > 1'.","section":"Abstract (wording)"}],"recommendation":"uncertain","confidential_remarks":"This review is based solely on the abstract; the full manuscript was not made available. The recommendation is therefore necessarily uncertain. I would need the theorem statements, definitions, and proof sketches to assess correctness and significance. If the full text was intended to be part of the review, it may not have been transmitted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You asked about 2508.10484. Since we only have the abstract, my take is provisional, but here it is.\n\nWhat is genuinely new: the w-coprime S-integer and S-integral ideal counting in positive characteristic is a natural next step in the established function-field program of counting squarefree, k-free, and coprime tuples. If the proofs are correct, you get explicit densities with Euler-product main terms, which would be useful to anyone working in arithmetic statistics over function fields. The announced method—analytic sieving plus Riemann-Roch and the Weil bound—is the standard toolkit, and in the usual fixed-genus, fixed-q setting it does give precise asymptotics. So nothing in the abstract is a red flag.\n\nWhat I cannot check: exact theorem statements, error-term regimes, the definition of \"w-coprime,\" and the associated local factors. The load-bearing point is uniformity. If the Weil bound is used for character sums over a range where the genus grows or S contains degree-1 primes, the error terms could compete with the main term. The abstract does not specify these parameter regimes. That is not a flaw in the work, just an unknown.\n\nThe abstract also doesn't compare with earlier counting results, so I cannot tell how much overlap there is with the authors' own past work or with other people's. There is no visible circularity or fitted normalization, but that is for lack of detail.\n\nIf the full text delivers rigorous asymptotics with explicit regimes, this is a solid contribution. I would want to see the uniformity arguments before trusting the error terms. For now, the paper deserves a serious referee rather than a desk reject: the question is whether the proof meets the standard. If the full text has theorem statements and proofs, send it to review. If it is just a sketch, the editor should ask to see the technical core first.","headline":"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.","tokens_in":1442,"tokens_out":1737,"would_cite":false,"duration_ms":20984,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R58","11G20","11N45"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["w-coprime","S-integers","S-integral ideals","function fields","finite fields","asymptotic counting","Riemann-Roch","Weil bound"],"falsifier":"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.","tokens_in":605,"feed_emoji":"🔢","tokens_out":9170,"duration_ms":96698,"temperature":0.7,"pith_summary":"This paper claims to derive asymptotic formulas, with explicit error terms, for the number of w-coprime S-integers and S-integral ideals of a given degree in an algebraic function field over a finite field, where w is any integer greater than 1 and S is a finite nonempty set of primes. The w-coprime condition is a natural high-weight generalization of squarefreeness: an object is counted only when no prime divides it with exponent at least w. The result matters because it converts a hard-looking divisibility counting problem into a main term that is a product of local factors, with the error term dictated by the Weil bound (the Riemann hypothesis for curves over finite fields) and Riemann-Roch counts. A sympathetic reader would take the paper to establish that the density of w-coprime objects in the S-integer ring exists and is computed by local Euler factors, with a discrepancy no larger than the square-root of the main term up to polynomial factors.","feed_headline":"Asymptotic formulas count w-coprime S-integers","feed_subtitle":"Riemann-Roch and the Weil bound control the error in function fields over finite fields.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[],"fun_headline_variants":["Explicit asymptotics for w-coprime S-integers","Riemann-Roch and Weil bound count w-coprime ideals","w-coprime S-integers: constant times q^n","Sieve in function fields yields exact counts"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit asymptotics for w-coprime S-integers","Riemann-Roch and Weil bound count w-coprime ideals","w-coprime S-integers: constant times q^n","Sieve in function fields yields exact counts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1147,"prompt_tokens":605,"completion_tokens":542,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":349,"completion_tokens_details":{"reasoning_tokens":472}},"tokens_in":349,"tokens_out":542,"duration_ms":6087,"temperature":1.0,"reasoning_tokens":472,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:24:10.192172+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}