Random Multiplicative Functions and Making Squares from Polynomial Values

Max Wenqiang Xu; R\'egis de la Bret\`eche; Victor Y. Wang

REVIEW 1 major objections 8 minor 34 references

Random Multiplicative Functions and Making Squares from Polynomial Values

T0 review · 1 major / 8 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Gaussian limits proven for random multiplicative functions on polynomial values

desk verdict New CLTs for Rademacher and extended Rademacher RMFs along polynomial values, with sharp Diophantine estimates for the quadratic case read the letter →

arxiv 2607.06398 v1 pith:SWNPJI4A submitted 2026-07-07 math.NT math.PR

Régis de la Bretèche , Victor Y. Wang , Max Wenqiang Xu This is my paper

classification math.NTmath.PR

keywords functionsmultiplicativerademacherachievecentralcountingequationsestablish

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

This paper proves that when you evaluate a polynomial P(n) at integers n = 1, 2, ..., N and feed those values into a random multiplicative function f (where f assigns random ±1 signs to each prime and extends multiplicatively), the partial sum Σ f(P(n)) behaves like a Gaussian random variable after normalization. The authors establish this central limit theorem for two types of random multiplicative functions — Rademacher (where f vanishes on non-square-free integers, modeling the Möbius function) and extended Rademacher (where f is defined completely multiplicatively). For the extended Rademacher case, the result is unconditional for all separable polynomials whose irreducible factors have degree at most 3. For the Rademacher case, the result holds under the assumption that a positive proportion of P(n) are square-free, which is known unconditionally in the same degree-at-most-3 setting. The core difficulty is not probabilistic but Diophantine: proving the central limit theorem reduces, via martingale methods, to showing that the equation P(n₁)P(n₂)P(n₃)P(n₄) = square has surprisingly few solutions where the nᵢ are all distinct — a paucity phenomenon. The paper establishes this paucity by combining tools from Diophantine geometry (bounds on integral points due to Evertse–Silverman and Reuss), results on Pell–Fermat equations (for the quadratic case, where the estimates are sharpest), and new combinatorial decompositions based on the complete graph K₄ that organize information about pairwise greatest common divisors of the P(nᵢ).

What carries the argument

The proof reduces the probabilistic central limit theorem to Diophantine counting via the McLeish martingale CLT. The key Diophantine tools are: (1) Hooley's results on the distribution of fundamental solutions to Pell–Fermat equations, used to bound off-diagonal solutions when deg P = 2; (2) Evertse–Silverman uniform bounds on integral points of superelliptic curves y² = d·P(x), used when deg P ≥ 3; (3) Reuss's determinant-method estimates for counting power-free values of cubic polynomials, used to control the set of n where P(n) has large square divisors (condition H_δ); (4) a combinatorial decomposition of the solution set based on labeling the six edges of the complete graph K₄ with gcd

What would settle it

A counterexample would be a separable polynomial P with all irreducible factors of degree ≤ 3 for which the normalized sum (1/√N) Σ g(P(n)) does not converge to N(0,1), or for which the off-diagonal solutions to P(n₁)P(n₂)P(n₃)P(n₄) = square are not o(N²).

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Extended reading notes

Core claim

The central discovery is a paucity phenomenon for the Diophantine equation P(n₁)P(n₂)P(n₃)P(n₄) = square: among all quadruples (n₁, n₂, n₃, n₄) with 1 ≤ nᵢ ≤ N, the number of solutions where all four nᵢ are distinct is o(N²), which is small enough to confirm Gaussian behavior of the partial sums. For degree-2 polynomials, the bound is O(N^{3/2+ε}), proved using the multiplicative structure of quadratic norm forms and growth estimates for solutions of Pell–Fermat equations. For degree ≥ 3 (with irreducible factors of degree ≤ 3), the bound is O(N^{14/9+ε}), proved using Evertse–Silverman uniform bounds on integral points, Reuss's determinant method for power-free values, and a graph-theoretic

Load-bearing premise

The Rademacher case (Theorem 1.1) assumes that a positive proportion of polynomial values P(n) are square-free. This is known unconditionally only when every irreducible factor of P has degree ≤ 3; for general polynomials it follows from the ABC Conjecture or the Square-Free Sieve Conjecture.

Editorial extensions

If this is right

The unconditional extended Rademacher CLT for polynomials with irreducible factors of degree ≤ 3 confirms the Gaussian prediction of Chinis–Shala for a broad class of polynomials, extending their degree-2 result.
The paucity estimates for P(n₁)P(n₂)P(n₃)P(n₄) = square are of independent Diophantine interest and may find applications in analytic number theory beyond random multiplicative functions.
The function-field analog of the square-free sieve condition (H_δ) is known unconditionally for arbitrary polynomials, suggesting that a polynomial Rademacher CLT over function fields F_q(t) may now be obtainable.
For specific polynomial families like P(X) = b²Q(X)⁴ − 1, the variance computation simplifies dramatically (the off-diagonal count vanishes identically by Bennett–Walsh), pointing to special polynomial structures where the Diophantine obstruction is minimal.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper proves two new CLTs: a Rademacher CLT for general separable P (Theorem 1.1, conditional on square-free density) and the first extended Rademacher CLT for polynomial values (Theorem 1.2, unconditional when all irreducible factors have degree ≤ 3). The Rademacher case for deg P = 2 overlaps with Chinis–Shala, but the extension to higher degree and the entire extended Rademacher treatment are new. The Diophantine estimates — Proposition 5.1 (variance, quadratic case, sharp at O(√N)) and Proposition 6.1 (fourth moment, quadratic case, O(N^{3/2+ε})) — are independently interesting and appear to be the sharpest available for these equations. The K₄-based combinatorial decomposition in §6 is a genuine idea: labeling edges of K₄ with gcd(P(nᵢ), P(nⱼ)) and optimizing the resulting exponent bounds is not a routine application of existing tools. The Z3 verification of E ≤ 3/2 in Lemma 6.3 is a legitimate use of machine assistance — the code is short, publicly available, and the optimization is a linear program that is independently checkable. The Pell–Fermat argument in Proposition 5.1 is intricate but sound; the factorization (14) and the case split by sizes of d and n₁+m₁√d correctly yield the √N bound, matching the lower bound (12). The stress-test note flags a typo in Lemma 3.1's hypothesis (a,b ≤ ε⁻¹N^{ε−1} should presumably be ε⁻¹N^{1+ε}). That is correct — it is a typo, and the underlying Bombieri–Pila bound is uniform in coefficients, so the applications go through regardless. The restriction to irreducible factors of degree ≤ 3 for the unconditional extended Rademacher case is a real limitation, but it is honestly discussed and the boundary is identified: extending to degree ≥ 4 requires a power-saving Square-Free Sieve Conjecture (H_δ), which nobody currently knows how to prove. The self-citations to Wang–Xu and Soundararajan–Xu provide the Steinhaus-case framework and are used appropriately. This is a solid paper by established researchers. It deserves a serious referee who can check the Pell equation argument in §5 and the combinatorial optimization in §6 line by line.

Referee Report

1 major / 8 minor

Summary. The paper proves central limit theorems for sums of Rademacher and extended Rademacher random multiplicative functions evaluated at polynomial values P(n). The main results (Theorems 1.1 and 1.2) confirm Gaussian limiting distributions for separable P with deg P >= 2, positive leading coefficient, and all irreducible factors of degree <= 3. The Rademacher case (Theorem 1.1) additionally assumes a positive density of square-free values (condition (2)), while the extended Rademacher case (Theorem 1.2) is unconditional. The proofs reduce, via the McLeish martingale CLT (following Soundararajan-Xu), to Diophantine counting problems: paucity of solutions to P(n_1)P(n_2) = square and P(n_1)P(n_2)P(n_3)P(n_4) = square. For deg P = 2, the authors develop a new method using Pell-Fermat equations (Propositions 5.1, 6.1). For deg P >= 3, they use Evertse-Silverman, Reuss's determinant method, and Bombieri-Pila (Propositions 8.1, 8.2, 3.6). A key optimization in Lemma 6.3 is verified using the Z3 SMT solver with publicly available code.

Significance. The paper makes substantial contributions to the study of random multiplicative functions along polynomial phases. The extended Rademacher CLT (Theorem 1.2) is unconditional and confirms the Gaussian prediction of Chinis-Shala for a broad class of polynomials, going beyond the Steinhaus case previously settled by Klurman-Shkredov-Xu. The Diophantine estimates are of independent interest: Proposition 5.1 gives a sharp O(sqrt(N)) bound for the variance problem in the quadratic case via Pell-Fermat theory, and Proposition 6.1 gives the key fourth-moment bound O(N^{3/2+epsilon}). The Z3-verified optimization in Lemma 6.3 (showing E <= 3/2) is independently checkable, and the code is publicly available. The K_4-based combinatorial decomposition of the fourth-moment counting problem is a novel and interesting technical contribution. The restriction to irreducible factors of degree <= 3 is clearly tied to the available Diophantine tools (Reuss, Evertse-Silverman) and is honestly disclosed.

major comments (1)

Lemma 3.1 (p. 7): The hypothesis states that the coefficients of P have absolute value at most epsilon^{-1} N^{epsilon - 1}, and similarly for a, b. For fixed epsilon > 0 and large N, the bound epsilon^{-1} N^{epsilon - 1} tends to 0, which is unsatisfiable for integers a, b >= 1. This is almost certainly a typo for epsilon^{-1} N^{1 + epsilon} (or similar). Since Lemma 3.1 is used in Proposition 8.1 (p. 19) and Proposition 8.2 (p. 20) to bound solutions to aP(n_1) = bP(n_2), the correct hypothesis needs to be stated. The proof itself (via Bombieri-Pila) is standard and the bound is correct once the hypothesis is fixed, but the statement as written is technically vacuous.

minor comments (8)

p. 7, Lemma 3.3: The remark mentions 'Gemini 3.1 Pro suggested to us in an incorrectly stated form' and 'GPT-5.5 Pro simplified the proof.' While acknowledging AI assistance is fine, the specific naming of AI models and their roles is unusual for a mathematics journal and could be streamlined.
p. 8, Proposition 3.6: The constraint delta_0 < 1/13 arises from the interplay between the three ranges I_1, I_2, I_3. The choice of theta requires delta_0 < theta < min(1/11, (1 - 11*delta_0)/2). It would help the reader to explicitly verify that (1 - 11*delta_0)/2 > delta_0 when delta_0 < 1/13, ensuring the interval for theta is non-empty.
p. 11, proof of Proposition 5.1: The factorization (14) and the subsequent case split are intricate. In the line after (15), 'we deduced << |Delta| N' should read 'we deduce << |Delta| N' for grammatical consistency.
p. 14, Lemma 6.2: The bound (18) involves a factor d^epsilon. It would be helpful to clarify whether this is d^epsilon or (log d)^{O(1)}, as the proof uses 2^{omega(d)} which is bounded by d^epsilon but also by tau(d).
p. 16, Lemma 6.3: The three bounds R, S_{ijk}, T_{ij} are defined in (23)-(27). The third bound T_{ij} uses the notation '2 - R + (beta_i + beta_j)/2'. In the proof, this is explained for (i,j) = (4,3) as '2 - (beta_1 + beta_2)/2 = 2 - R + (beta_4 + beta_3)/2'. This is correct but the general formula could be stated more explicitly for arbitrary (i,j).
p. 19, Proposition 8.1: The bound M_4(N, A) << A^{11/9} N^{14/9 + epsilon} is stated for A <= N^5. The proof takes N^delta = A^{1/9} N^{4/9}, which requires A <= N^5 for delta <= 1. This is consistent but the role of the exponent 5 could be flagged more prominently.
p. 22, Appendix A: The Z3 code is clearly presented. The sharpness check output gives specific values of alpha variables. It would be useful to include a brief human-readable verification that these values indeed give E = 3/2, to complement the machine check.
References: The arXiv preprint [5] (Bhargava) and [2] (Alpoge) are cited with arXiv numbers. If published versions exist, they should be updated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment. The referee raises one major comment, which we address below.

read point-by-point responses

Referee: Lemma 3.1 (p. 7): The hypothesis states that the coefficients of P have absolute value at most ε^{-1} N^{ε - 1}, and similarly for a, b. For fixed ε > 0 and large N, the bound ε^{-1} N^{ε - 1} tends to 0, which is unsatisfiable for integers a, b >= 1. This is almost certainly a typo for ε^{-1} N^{1 + ε} (or similar). Since Lemma 3.1 is used in Proposition 8.1 (p. 19) and Proposition 8.2 (p. 20) to bound solutions to aP(n_1) = bP(n_2), the correct hypothesis needs to be stated. The proof itself (via Bombieri-Pila) is standard and the bound is correct once the hypothesis is fixed, but the statement as written is technically vacuous.

Authors: The referee is entirely correct. The bound ε^{-1} N^{ε-1} in the hypothesis of Lemma 3.1 is a typo; it should read ε^{-1} N^{1+ε}. With the corrected hypothesis, the condition is satisfiable for integers a, b ≥ 1 when N is large, and the proof via Bombieri–Pila goes through unchanged. We have verified that in all applications of Lemma 3.1—specifically in Propositions 8.1 and 8.2, and in the third bound of Lemma 6.3—the parameters a, b arise as products of square-free parts and their divisors, which are indeed bounded by N^{1+ε} (up to N^ε factors absorbed by the ε in the exponent). The polynomial P is fixed throughout, so the condition on its coefficients is automatically satisfied for large N. We will correct the statement in the revised manuscript. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found; the derivation chain is self-contained against external benchmarks.

full rationale

The paper's central claims (Theorems 1.1 and 1.2) are derived from a chain of Diophantine estimates and a standard CLT framework (McLeish martingale CLT via Theorem 2.1/2.2). The key Diophantine inputs—Hooley's results on Pell equations (Proposition 5.1), Evertse–Silverman (Lemma 3.3), Reuss's determinant method (Proposition 3.6), and Bombieri–Pila (Lemma 3.1)—are all externally sourced results, not self-citations. The self-citations present ([34] Wang–Xu, [32] Soundararajan–Xu, [24] Klurman–Shkredov–Xu) provide the Steinhaus-case framework and paucity results that the present work extends, but the present results are not equivalent to those by construction. The Z3-verified optimization in Lemma 6.3 is independently checkable via the provided code repository. The CLT criteria in Theorem 2.1 are attributed to Soundararajan–Xu [32] but are a routine modification of a standard McLeish CLT application, not a self-serving definition. No step in the derivation chain reduces to its own inputs by construction.

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

No free parameters are fitted to data. No new mathematical entities (particles, forces, dimensions) are postulated. The axioms are all standard or domain-specific results from the prior literature, clearly cited. The main conditional assumption (H_δ / condition (2)) is explicitly stated as an open problem, not postulated as true.

assumptions (8)

standard math McLeish Martingale Central Limit Theorem (Theorem 2.1, §2): the standard probabilistic framework reducing CLT to moment conditions.
Invoked in §2 to reduce the probabilistic claim to Diophantine counting. Standard tool in the field.
domain assumption Huxley's bound on polynomial congruence roots (eq. 8, §3): ρ_P(q) ≤ |disc(P)|^{1/2} (deg P)^{ω(q)}.
Used throughout §3–8 to bound the number of solutions to polynomial congruences. External result [23].
domain assumption Evertse–Silverman uniform bound on integral points (Lemma 3.3, §3): #{(x,y) ∈ Z² : y² = dP(x)} ≤ 2^{O_P(1+ω(d))}.
Used in Lemma 3.3 and Proposition 8.1 to bound solutions to P(x) = dy². External result [12].
domain assumption Reuss determinant-method estimate (§3, Proposition 3.6): bounds on N(N;A,B) for cubic factors.
Used in Proposition 3.6 to establish H_δ for polynomials with irreducible factors of degree ≤ 3. External result [31].
domain assumption Condition (2): lim inf #{n ≤ N : μ²(P(n)) = 1}/N > 0 (positive density of square-free values).
Load-bearing assumption for Theorem 1.1 (Rademacher case). Known for deg ≤ 3 unconditionally; requires ABC or Square-Free Sieve Conjecture in general.
domain assumption H_δ (condition (9), §3): B(N, N^{1/2}) ≪ N^{1−δ} — power-saving Square-Free Sieve Conjecture.
Assumed in Theorem 8.3 for degree ≥ 4 extended Rademacher case. Proposition 3.6 proves H_δ for factors of degree ≤ 3.
standard math Bombieri–Pila bound on integral points on curves (Lemma 3.1, §3).
Used to bound solutions to aP(n₁) = bP(n₂). External result [8].
domain assumption Hooley's result on Pell equations (§5, cited as [21, Theorem 1]).
Used in Proposition 5.1 to bound S(D) and establish the variance estimate for quadratic P.

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Pith. "Pith review of Random Multiplicative Functions and Making Squares from Polynomial Values." pith.science (2026). https://pith.science/paper/SWNPJI4A

@misc{pith2026260706398,
  author       = {Pith},
  title        = {Pith review of: Random Multiplicative Functions and Making Squares from Polynomial Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWNPJI4A}},
  note         = {Machine review of arXiv:2607.06398}
}

abstract

For a large family of polynomials $P(X)\in \mathbb{Z}[X]$, we prove central limit theorems for $\sum_{n\le N} f(P(n))$ for both Rademacher and extended Rademacher multiplicative functions $f$. To achieve this, we establish a paucity phenomenon in counting solutions to \[P(n_1)P(n_2)P(n_3)P(n_4) = \square, \quad 1\le n_1, n_2, n_3, n_4 \le N.\] Results of Hooley, Evertse--Silverman, and Reuss play an important role in the proof. Our estimates are sharpest for $\deg P = 2$, thanks to the rich theory of Pell--Fermat equations.

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Works this paper leans on

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Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.

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Akhtari, The Diophantine equationaX 4 −bY 2 = 1,J

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work page 2009

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work page Pith review arXiv 2021

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[4] [4]

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