On a Conjecture of Erd\H{o}s over Function Fields

Likun Xie

REVIEW 1 major objections 4 minor 20 references

On a Conjecture of Erd\H{o}s over Function Fields

T0 review · 1 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Every residue class modulo a squarefree polynomial is a product of two monic irreducibles once the field is large enough.

desk verdict The claimed theorem is already Sawin's, and the new proof has a real gap: Λ-counted prime powers do not give the required irreducible pairs, so Theorem 1.1 doesn't follow as written. read the letter →

arxiv 2510.17612 v3 pith:QKDZHZV2 submitted 2025-10-20 math.NT math.AG

Likun Xie This is my paper · ORCID

classification math.NTmath.AG MSC 11T5511T24

keywords functionfieldsirreduciblepolynomialsresidueclassescharactersumsperversesheavesequidistributiontracefunctionslargeqregime

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

The paper proves a function-field version of a classical conjecture on products of primes: for any squarefree polynomial f of degree n, once the finite field F_q has q ≥ Q(n) with Q(n) of the form n^{κn}, every invertible residue class modulo f can be written as a product of two monic irreducible polynomials of degree at most n. The proof controls the error in the character-sum expression for the number of such representations, obtaining a uniform saving of q^{-1/2} against a main term of size q^n. This suffices to beat the main term for large q. An explicit threshold Q(n) = n^{23n} is given. The argument uses a one-dimensional perverse-sheaf construction rather than a higher-dimensional one, and is also shown to work for fixed degrees of the two factors.

What carries the argument

The engine of the proof is a perverse sheaf N on the multiplicative group, constructed from the L-function of a multiplicative character χ of the unit group of k[X]/(f). Its Tannakian monodromy group is the full general linear group GL(n-1), and for a 'good' multiplicative character ρ of k^×, the trace of a representation Λ at the Frobenius element of N tensored with a Kummer sheaf equals a Frobenius trace of the corresponding object in the Tannakian category. Since the object is non-punctual, the Riemann Hypothesis for the underlying pure weight -1 lisse sheaf gives a pointwise q^{-1/2} bound. A Frobenius character formula (expanding products of power sums into Schur functions) converts the

What would settle it

For some n, q with q ≥ Q(n), and a squarefree f of degree n, compute S(a;n) and enumerate all pairs of monic irreducibles of degree ≤ n for every residue class a. If some a satisfies S(a;n) > 0 yet no such pair exists, the theorem collapses. Concretely, if the only pairs of monic prime powers (g,h) with gh ≡ a (mod f) have g = p^k or h = p^ℓ with k,ℓ ≥ 2, then a is not a product of two irreducibles, and the proof's asserted bridge is broken.

Watch

Extended reading notes

Core claim

The central claim is that the counting error R(a,n) satisfies |R(a,n)| < M(a,n) for every residue class a, where M(a,n) ~ q^n is the expected number of ordered pairs of monic irreducibles of degree ≤ n with product congruent to a. The error is expressed as a sum over characters of products of traces of unitarized L-function matrices. For 'totally ramified generic' characters, the associated perverse sheaf has generic rank n, Tannakian dimension n-1, and arithmetic monodromy group GL(n-1); this forces the relevant Frobenius traces to be bounded by an explicit multiple of q^{-1/2}. Summing over the remaining characters with elementary bounds and a partition identity yields the final estimate.

Load-bearing premise

The proof shows that a weighted sum counting products of prime powers modulo f is positive, and asserts without a separate justification that this positivity forces the existence of a genuine product of two monic irreducibles; if that bridge fails, Theorem 1.1 does not follow from the argument.

Editorial extensions

If this is right

For every n≥2 there is an explicit threshold Q(n)=n^{23n} such that for all q≥Q(n) and every squarefree f of degree n, E(k,f)=B^× — every residue class is a product of two monic irreducibles of degree at most n.
The representation can be realized with fixed degrees: for any r with 1≤r<2n, every class is representable as f1·f2 with deg f1=r and deg f2=2n−r.
The same one-dimensional template gives a q^{-1/2} saving for more general sums over characters, for example when the von Mangoldt weights are replaced by the Möbius function or by divisor functions.
The error bound is explicit: |R(a,n)| is at most C(n) q^{n−1/2} plus a smaller term, with C(n)=(2n)^{2n+3} e^{π√(4n/3)}.

Reading between the lines

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

The one-dimensional construction suggests that any argument supported on a curve can at best yield a q^{-1/2} saving; matching the stronger q^{-n/2} cancellation of higher-dimensional approaches likely requires going beyond one-dimensional supports — a divide that appears structural rather than technical.
The explicit threshold Q(n)=n^{23n} is extremely conservative; sharper estimates for partition numbers or a different decomposition of the character sum could lower the exponent, potentially bringing the result closer to the range accessible by simpler analytic methods.
The paper's proof leaves a genuine logical gap between positivity of the weighted sum (which counts products of prime powers) and the existence of an actual pair of irreducibles; if that bridge is filled, the analytic estimates here would already imply the theorem, but until then the theorem rests on an unproven step.
The method may be adaptable to other moduli beyond squarefree f, but the proof's reliance on total ramification and generic characters suggests the error terms would degrade for non-squarefree or imprimitive characters.

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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

Two things you should know before reading this paper. First, the main theorem is not new — the author says so himself: Sawin proved the same statement with a much stronger error term. The contribution is supposed to be a one-dimensional proof and an explicit Q(n). Second, that proof doesn't actually work as written. The central object S(a;n) is a Λ-weighted character sum, and by orthogonality it counts pairs of monic prime powers (not just irreducibles) whose product is congruent to a mod f. The paper shows |R(a,n)| < M(a,n), which gives S(a;n) > 0. That only guarantees a pair of prime powers. The theorem requires two monic irreducibles. Section 2 says 'it suffices to show R(a,n)=o(M(a,n))' but that's false — it suffices only if you also know that the contribution from proper prime powers is negligible, and that argument is absent. This is a genuine derivation gap, not a stylistic issue.

The technical core is not rubbish. The perverse-sheaf estimates in §3 are plausible and fit Katz's convolution framework; the explicit bound on R(a,n) with C(n) is concrete; and the general template for other arithmetic weights (Möbius, divisor functions) is a nice idea. The constant Q(n)=n^{23n} is also honestly derived. But all of that only bounds a Λ-weighted sum.

The fix is probably elementary. Proper prime powers of degree ≤n are sparse — roughly O(n q^{n/2}) polynomials — and each residue class mod f has at most two monic representatives of a given degree, so the bad contribution should be o(M(a,n)). Adding a lemma to that effect would close the gap. Without it, Theorem 1.1 does not follow.

Would I send it to a referee? Yes — the method is serious and the gap is likely repairable, and a good referee could push the author to finish it. But don't send it out as-is, and don't let anyone think the main result is new. It's a potentially useful technique in need of a missing lemma.

Referee Report

1 major / 4 minor

Summary. The paper claims to prove the function-field analogue of Erdős's conjecture in the large-q regime: for every n≥2 there is an explicit Q(n) (of shape n^{23n}) such that for every finite field F_q with q≥Q(n) and every squarefree f∈F_q[X] of degree n, every residue class modulo f is a product of two monic irreducible polynomials of degree at most n. The proof defines for each a∈B^× a von Mangoldt-weighted character sum S(a;n), uses Katz's Tannakian–equidistribution framework to show that the non-principal part R(a;n) is bounded by C(n) q^{n−1/2} + 2(n+1)n^4 q^{n−1}, while the principal part M(a;n) is at least q^n−O(n^2), so |R(a;n)|<M(a;n) for q sufficiently large. The paper then concludes that every residue class is a product of two monic irreducibles. An additional remark claims a variant with fixed degrees r and 2n−r.

Significance. If the advertised conclusion were established, the paper would be a valuable complement to Sawin's stronger theorem [19]: it gives a simpler one-dimensional argument, an explicit large-q threshold, and illustrates how Katz's convolution–equidistribution framework yields a natural q^{-1/2} saving. The technical core—constructing the sheaf N(λ,χ), proving the Lie-irreducibility and Tannakian groups GL(n−1), and extracting the character-sum estimates via partition expansions—is substantial and appears internally coherent, assuming the quoted theorems of Katz. The explicit constant κ=23 for Q(n) is also supplied. However, the conclusion as written does not follow from the estimates, because the sums count prime powers rather than irreducibles.

major comments (1)

[Section 2, 'To establish Theorem 1.1…' and final step of Section 3] The central implication is missing and load-bearing. S(a;n) is defined using the von Mangoldt function Λ, so S(a;n)>0 (which is what |R(a;n)|<M(a;n) gives) implies only the existence of monic g,h with Λ(g)Λ(h)>0, i.e. each of g,h is a power of a single monic irreducible. Theorem 1.1 requires the factors to be irreducible (exponent 1). No argument in §2–§4 bounds or removes pairs in which at least one factor is a proper prime power (k≥2). Thus positivity of S(a;n) does not imply E(k,f)=B^×. This is exactly the step bridging the analytic estimates and the advertised number-theoretic conclusion, and it is absent. A likely elementary lemma estimating the contribution of proper prime powers (e.g. O(n^2 q^{n/2})) would close the gap, but it must be stated and proved.

minor comments (4)

[Section 2, definition of S(a;d)] The sentence 'one may restrict to degg=d since the contribution is dominated by polynomials of the highest degree' is not used and is inconsistent with the later sums over all i,j≤n and the main term M(a;n) containing all k≤n. Since the actual proof sums over all degrees, this remark is misleading and should be removed or clarified.
[Section 2, notation] The expression 'R(a,n)=o_n(M(a,n))' is not a proper asymptotic in n alone: q is the varying parameter and n is fixed. The intended meaning is |R(a,n)|<M(a,n) for q≥Q(n). Please state this as an explicit inequality, as done later in the section.
[Proof of Proposition 3.3] The inequality 'If √q≥2n+1, then |Bad|≤√q−1' should cite Theorem 3.2(1) (at most 2n bad characters) to justify the deduction; as written the reader must supply that link.
[Final remark on E′(k,f)] The variant with fixed degrees r and 2n−r suffers from the same prime-power gap: the estimates imply existence of a pair of prime powers, not irreducibles. This should be addressed together with the main theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper relies on external equidistribution theorems and standard Weil/Deligne bounds; the flagged concern is a derivation gap, not a circular reduction.

full rationale

The paper does not assume Theorem 1.1 or any equivalent formulation. The main estimate for R(a,n) is derived from Katz's convolution–equidistribution framework ([12,13]) and Weil's Riemann Hypothesis, and the final comparison |R(a,n)| < M(a,n) is a quantitative inequality with an explicit Q(n). There is no fitted parameter renamed as a prediction: M(a,d) is computed from the prime polynomial theorem, and the claimed saving is unconditional once q is large. The introductory use of Sawin's Lemma 9.14 is an independent external result, not a self-citation by the author, and the main proof does not depend on it. The skeptical concern that positivity of the Λ-weighted sum S(a;n) yields prime-power pairs rather than irreducible pairs is a genuine derivation gap: the text asserts 'To establish Theorem 1.1, it suffices to show ... R(a,n)=o_n(M(a,n))' without proving that positive Λ-counts can be converted into monic irreducible factors. However, that is a missing implication in the proof, not circularity: no equation defining S(a;n), M(a,n), or R(a,n) is equivalent by construction to the target statement E(k,f)=B^×. No load-bearing step reduces to its own input by definition, and there is no self-citation chain forcing the conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof is built almost entirely on external theorems: Weil's RH, Deligne's weight arguments, Katz's Tannakian/equidistribution results and counting lemmas. No new constants are fitted to data; Q(n) is obtained from explicit inequalities. No new entities are postulated.

assumptions (5)

standard math Weil's Riemann Hypothesis for function fields: reciprocal roots of L(χ,T) have absolute value 1 or q^{1/2}.
Used throughout, especially (2.4) and Proposition 3.3, to bound Ψ(m,χ) and Frobenius traces.
domain assumption Katz's Theorem 5.1 giving Tannakian groups Ggeom=Garith=GL(n-1) for N(λ,χ) for generic characters.
External deep theorem on which Proposition 3.3 and the q^{-1/2} saving rest.
domain assumption Counting bounds for totally ramified generic characters from Katz's Lemmas 6.4–6.6.
Used in Corollary 3.5 to split the character sum into generic and non-generic parts.
standard math Deligne's Riemann Hypothesis for weights and purity, and the Grothendieck–Lefschetz trace formula.
Used in Proposition 3.3 to bound the Frobenius trace on perverse sheaves.
standard math Prime polynomial theorem for F_q[t] giving Σ_{deg g=d} Λ(g)=q^d.
Used at (2.1) to compute the main term M(a,d).

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Cite this review

Pith. "Pith review of On a Conjecture of Erd\H{o}s over Function Fields." pith.science (2026). https://pith.science/paper/QKDZHZV2

@misc{pith2026251017612,
  author       = {Pith},
  title        = {Pith review of: On a Conjecture of Erd\Hos over Function Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKDZHZV2}},
  note         = {Machine review of arXiv:2510.17612}
}

abstract

Using Katz's equidistribution framework, we show that for any squarefree polynomial $f \in \mathbb{F}_q[t]$ of degree $n \ge 2$, every residue class modulo $f$ can be represented as a product of two monic irreducible polynomials of degree at most $n$, provided $q$ is sufficiently large in terms of $n$. This gives the function-field analogue of a conjecture of Erd\H{o}s in the large-$q$ regime. Sawin previously proved this representation with stronger square-root cancellation via a higher-dimensional sheaf-theoretic construction. This note presents a one-dimensional argument that yields a natural $q^{-1/2}$ saving.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 1 linked inside Pith

[19]

Sawin, W. (2024). Square-root cancellation for sums of factorization functions over squarefree progressions in Fq[t]. Acta Mathematica, 233(2), 285–418

2024
[1]

Apostol, T. M. (2013).Introduction to analytic number theory. Springer Science & Business Media

2013
[2]

(2003).Representations of compact Lie groups (Vol

Bröcker, T., & Tom Dieck, T. (2003).Representations of compact Lie groups (Vol. 98). Springer Science & Business Media

2003
[3]

(2012).Néron models (Vol

Bosch, S., Lütkebohmert, W., & Raynaud, M. (2012).Néron models (Vol. 21). Springer Science & Business Media

2012
[4]

Chen, J.-R. (1973). On the representation of a larger even integer as the sum of a prime and the product of at most two primes.Scientia Sinica, 16(2), 157–176

1973
[5]

Deligne, P. (1980). La conjecture de Weil: II.Publications Mathématiques de l’IHÉS, 52, 137-252

1980
[6]

Deligne, P., & Milne, J. S. (1982). Tannakian categories. In Hodge cycles, motives, and Shimura varieties (pp. 101-228). Berlin, Heidelberg: Springer Berlin Heidelberg

1982
[7]

M., & Sárközy, A

Erdős, P., Odlyzko, A. M., & Sárközy, A. (1987). On the residues of products of prime numbers.Periodica Mathematica Hungarica, 18(3), 229-239

1987

Show all 20 references

[8]

Forey, A., Fresán, J., & Kowalski, E. (2021). Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution. arXiv preprint arXiv:2109.11961

2021 arXiv
[9]

(2013).Representation theory: a first course (Vol

Fulton, W., & Harris, J. (2013).Representation theory: a first course (Vol. 129). Springer Science & Business Media

2013
[10]

(ed.),Cohomologie ℓ-adique et fonctionsL, Lecture Notes in Mathematics, vol

Illusie, L. (ed.),Cohomologie ℓ-adique et fonctionsL, Lecture Notes in Mathematics, vol. 589, Springer–Verlag, Berlin–New York, 1977. (Séminaire de Géométrie Algébrique du Bois-Marie 1965–1966 [SGA 5].)

1977
[11]

Katz, N. M. (1996).Rigid local systems (Annals of Mathematics Studies).Princeton University Press

1996
[12]

Katz, N. M. (2012). Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms. Princeton University Press

2012
[13]

Katz, N. M. (2013). On a question of Keating and Rudnick about primitive Dirichlet characters with squarefree conductor. Int. Math. Res. Not. IMRN, 14, 3221-3249

2013
[14]

P., & Rudnick, Z

Keating, J. P., & Rudnick, Z. (2014). The variance of the number of prime polynomials in short intervals and in residue classes.International Mathematics Research Notices, 2014(1), 259-288

2014
[15]

& Teräväinen, J

Matomäki, K. & Teräväinen, J. (2024). Products of primes in arithmetic progressions.Journal für die reine und angewandte Mathematik (Crelles Journal), 2024(808), 193-240

2024
[16]

Milne, J. S. (2017).Algebraic groups: the theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press

2017
[17]

(2013).Number theory in function fields(Vol

Rosen, M. (2013).Number theory in function fields(Vol. 210). Springer Science & Business Media

2013
[18]

Sawin, W., & Shusterman, M. (2022). On the Chowla and twin primes conjectures overFq[t]. Annals of Math- ematics, 196(2), 457-506

2022
[20]

(1948).Sur les courbes algébriques et les variétés qui s’en déduisent(Actualités Scientifiques et Indus- trielles, No

Weil, A. (1948).Sur les courbes algébriques et les variétés qui s’en déduisent(Actualités Scientifiques et Indus- trielles, No. 1041; Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. 7). Hermann. Max-Planck-Institut für Mathematik Vivatsgasse 7, 53...

1948

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [19]

Sawin, W. (2024). Square-root cancellation for sums of factorization functions over squarefree progressions in Fq[t]. Acta Mathematica, 233(2), 285–418

2024

[2] [1]

Apostol, T. M. (2013).Introduction to analytic number theory. Springer Science & Business Media

2013

[3] [2]

(2003).Representations of compact Lie groups (Vol

Bröcker, T., & Tom Dieck, T. (2003).Representations of compact Lie groups (Vol. 98). Springer Science & Business Media

2003

[4] [3]

(2012).Néron models (Vol

Bosch, S., Lütkebohmert, W., & Raynaud, M. (2012).Néron models (Vol. 21). Springer Science & Business Media

2012

[5] [4]

Chen, J.-R. (1973). On the representation of a larger even integer as the sum of a prime and the product of at most two primes.Scientia Sinica, 16(2), 157–176

1973

[6] [5]

Deligne, P. (1980). La conjecture de Weil: II.Publications Mathématiques de l’IHÉS, 52, 137-252

1980

[7] [6]

Deligne, P., & Milne, J. S. (1982). Tannakian categories. In Hodge cycles, motives, and Shimura varieties (pp. 101-228). Berlin, Heidelberg: Springer Berlin Heidelberg

1982

[8] [7]

M., & Sárközy, A

Erdős, P., Odlyzko, A. M., & Sárközy, A. (1987). On the residues of products of prime numbers.Periodica Mathematica Hungarica, 18(3), 229-239

1987