REVIEW 2 major objections 4 minor 9 references
On the Fractional Parts of Polynomials Modulo $p$
T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read The paper proves an exact formula for a quadratic residue counting function and uses it to refute Sun's conjectured O(√p) error bound.
desk verdict A real paper with one clean new result — exact class-number evaluation for the quadratic case and unconditional sqrt(p) log log p lower bound — and a load-bearing lemma that needs a cleaner statement before acceptance. 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 proof uses finite Fourier expansions to express the indicator function of the upper half-interval as an exponential sum, converting the counting problem into estimates of incomplete Gauss sums. For general polynomials the Weil bound controls the complete sums; for quadratics an explicit incomplete-Gauss-sum bound (due to Korolev) removes one logarithm; for symmetric polynomials the half-interval sum is completed to a full exponential sum via an involution. For monomials x^m, the count is rewritten using Dirichlet characters restricted to the upper half of the field, and the error term becomes a sum of incomplete character sums whose GRH-conditional bound is O(√p log log p). The exact qua
What would settle it
For a sequence of large primes p ≡ 3 (mod 8), compute f₂(p) directly by brute force and compare with (p−1)/4 − (3/2)h(−p) (with h(−p) computed independently). If the difference is nonzero for any p > 3, the exact formula is false. More decisively, if for all primes up to some large X the ratio |f₂(p) − p/4| / √p remains bounded (e.g., by a constant less than 1), then the claimed infinite subsequence with ≫ √p log log p does not exist and the lower bound is wrong.
Extended reading notes
Core claim
The central discovery is Theorem 1.6: for any odd prime p > 3, the number f₂(p) of k in the half-interval 1 ≤ k < p/2 with {k²/p} > 1/2 is exactly (p−1)/4 when p ≡ 1 mod 4, (p−1)/4 − (3/2)h(−p) when p ≡ 3 mod 8, and (p−1)/4 − (1/2)h(−p) when p ≡ 7 mod 8, where h(−p) is the class number. Combining this with Dirichlet's class number formula and an unconditional lower bound for L(1, χ_p), the authors obtain |f₂(p) − p/4| ≫ √p log log p for infinitely many primes p, refuting Sun's conjectured O(√p) error. They further show that, assuming GRH, the upper bound O_m(√p log log p) holds for every even monomial x^m, so the log log p factor is essentially the right order in the quadratic case.
Load-bearing premise
The unconditional lower bound (Lemma 2.6) asserts that L(1, χ_p) > c log log p for infinitely many primes p ≡ 3 mod 4; if this 1950 theorem does not apply to prime moduli in exactly this form, the claimed refutation of Sun's O(√p) conjecture collapses.
Editorial extensions
If this is right
- Sun's conjecture that f_m(p) = p/4 + O_m(√p) is false at least for m = 2; the true error is often as large as √p log log p.
- The exact formula gives a deterministic O(p) algorithm for computing f₂(p) once h(−p) is known, and conversely the count can be used to compute class numbers in this range.
- The matching GRH-conditional upper bound for all even monomials indicates that the log log p factor is a genuine feature of the half-interval distribution, not an artifact of the proof.
- The refined O(√p log p) bound for symmetric polynomials shows that reflection symmetry is a sufficient structural condition for removing one logarithm, suggesting that other group symmetries (as in Theorem 1.8) behave similarly.
Reading between the lines
- One testable extension is to check the exact formula computationally for primes up to a few thousand; any deviation would not only disprove Theorem 1.6 but also reveal a flaw in the class-number relation, a stronger check than the asymptotics.
- The lower-bound argument likely carries over to other quadratic forms (e.g., φ(x)=ax²+bx+c) by similar class-number identities, which would extend the refutation of Sun's conjecture to a family of polynomials, not just the monomial.
- The connection to L(1, χ) suggests that the true distribution of f₂(p) is governed by the same fluctuations that give Littlewood's bounds, so under GRH one expects the log log p lower bound to hold for almost all primes, not just a sparse subsequence.
- The method of using a fundamental domain for the d-th roots of unity (Theorem 1.8) may generalize to counting functions on higher-dimensional tori, where Fourier expansion and Weil-type bounds remain effective.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies a finite analogue of Weyl equidistribution: for a fixed polynomial φ over Z and an odd prime p, it counts k in the lower half-interval I={1,...,(p-1)/2} for which the fractional part of φ(k)/p lies in the upper half J={(p+1)/2,...,p-1}. The main unconditional result (Theorem 1.1) is f_φ(p)=p/4+O_φ(√p log^2 p), obtained by expanding the indicator of J in finite Fourier series and applying the Weil bound to the complete sums. For quadratic polynomials and for polynomials with a reflection symmetry φ(c−x)=φ(x), the error is improved to O_φ(√p log p). For even monomials x^m, a character-sum argument under GRH yields O_m(√p log log p) (Theorem 1.7). The central result is Theorem 1.6, an exact class-number formula for m=2: f_2(p)=(p-1)/4 for p≡1 mod4, and (p-1)/4 - (3/2)h(-p) or (p-1)/4 - (1/2)h(-p) for p≡3 mod8 or p≡7 mod8. Combined with the class-number formula and an Ω-result for L(1,χ_p), this gives |f_2(p)-p/4|≫√p log log p infinitely often, disproving Sun's conjectured O(√p) error term. A final section generalizes the method to a fundamental domain for the d-th roots of unity.
Significance. Assuming the lower-bound input is correctly quoted, the paper settles the quadratic case of Sun's 2018 problem: the error is not O(√p), and the true upper order is √p log log p under GRH, matching the unconditional lower bound. The exact evaluation in Theorem 1.6 is a clean and valuable result, and the Fourier/character-sum framework is natural and clearly presented. The arguments are internally consistent; all external tools (Weil, Korolev, Montgomery–Vaughan, BCE) are standard and explicitly cited. The paper contains no fitted parameters and no circular reasoning, and the main claims are falsifiable. The novelty is incremental in method but decisive for the specific conjecture.
major comments (2)
- [Lemma 2.6; Proof of Theorem 1.6 (Section 4)] The unconditional lower bound, and hence the disproof of Sun's conjecture, rests on Lemma 2.6. As stated, Lemma 2.6 is quoted from [1, Theorem 1(C)] but its original hypotheses are not given; the authors should either reproduce the statement from [1] or prove the specialization to primes p≡3 mod4. In addition, the proof passes from h(-p)=√p/π L(1,χ_K) to the lower bound by invoking Lemma 2.6, which is stated for L(1,χ_p) with χ_p(n)=(n/p). The equality of χ_K (the character of Q(√-p)) and χ_p for p≡3 mod4 is true but is not stated; without it the application is not justified. Since this is the load-bearing step for the central claim, it must be fixed.
- [Lemma 2.3; Proof of Theorem 1.3 (Section 3)] The O_φ(√p log p) quadratic bound uses max_{a≠0}|S(a)| and asserts that Lemma 2.3 gives a bound uniform in a. Lemma 2.3 is stated for one fixed quadratic polynomial f(x)=Ax^2+Bx+C with p∤A; the text does not explicitly state that the constant 7.0508 is independent of the coefficients A,B,C (and hence of the multiplier a). The authors should quote the precise theorem from [3] or [2] guaranteeing uniformity over all coefficients (or over a∈F_p^*) so that the max over a is justified.
minor comments (4)
- [Section 4, Proof of Theorem 1.6] The sentence 'which counts the quadratic residues of elements in J belonging to I' is confusing; it should read 'which equals the number of quadratic residues in J'.
- [Abstract and Remark 4.1] The phrase 'factor log log p is best possible' should be qualified as conditional on GRH, since the only matching upper bound cited (Littlewood) is GRH-conditional. The unconditional result alone shows only that the error is not O(√p).
- [Lemma 2.6] Please give a more precise reference for the quoted theorem, including page or theorem number from [1], and state explicitly whether the constant is effective. This would also help verify the prime-modulus specialization.
- [Proof of Theorem 1.1] The main term is first written as (p-1)^2/(4p) and later simplified to p/4+O(1). It may help the reader to state at the outset that the difference is absorbed into the error term.
Circularity Check
No significant circularity: the paper's derivations build on external standard estimates and the class number formula, not on its own conclusions.
full rationale
I walked the derivation chain from the Fourier-expansion setup through Theorems 1.1, 1.3, 1.4, 1.6, 1.7, and 1.8. The counting functions are converted to exponential sums or character sums using standard identities, and the bounds come from external lemmas: Weil's bound (Lemma 2.1), finite Fourier coefficient estimates (Lemma 2.2), Korolev's incomplete Gauss sum bound (Lemma 2.3), GRH character-sum bounds (Lemma 2.4), Washington's class-number identities (Lemma 2.5), and the Bateman–Chowla–Erdős lower bound for L(1,χ_p) (Lemma 2.6). The exact formula in Theorem 1.6 is derived algebraically from Lemma 2.5 and the Legendre-symbol rewriting; it is not an input to itself. The lower-bound improvement uses Lemma 2.6 as an external Ω-result, not as a consequence of the paper's own theorem. There are no fitted parameters, no self-citations carrying the argument, and no target formula assumed in the proof. While Lemma 2.6 is load-bearing for the unconditional lower bound, that is a matter of external correctness/verifiability, not circularity. The paper also makes its conditional results explicitly dependent on GRH rather than smuggling them in as consequences. Overall, no reduction of a claimed result to its own inputs was found.
Assumptions & free parameters
assumptions (8)
- standard math Weil bound for complete exponential sums over F_p (Lemma 2.1, [4] Thm 5.38)
- standard math L1 Fourier coefficient bound for interval indicators (Lemma 2.2)
- domain assumption Uniform 7.0508√p+2 bound for incomplete quadratic Gauss sums (Lemma 2.3, Korolev [3] / Cochrane [2])
- domain assumption GRH bound for incomplete Dirichlet character sums (Lemma 2.4, Montgomery–Vaughan [6] Thm 2)
- standard math Classical facts on quadratic residues in (0,p/2) and class number formula (Lemma 2.5, Washington [8])
- domain assumption Unconditional Ω-result L(1,χ_p) ≫ log log p for infinitely many primes p≡3 mod 4 (Lemma 2.6, Bateman–Chowla–Erdős [1])
- domain assumption GRH assumption as used in Theorem 1.7 and the Littlewood matching upper bound
- domain assumption Symmetry hypothesis φ(c−x)=φ(x) and fundamental-domain conditions in Theorems 1.4 and 1.8
Cite this review
Pith. "Pith review of On the Fractional Parts of Polynomials Modulo $p$." pith.science (2026). https://pith.science/paper/T46ZELFJ
@misc{pith2026260721259,
author = {Pith},
title = {Pith review of: On the Fractional Parts of Polynomials Modulo $p$},
year = {2026},
howpublished = {\url{https://pith.science/paper/T46ZELFJ}},
note = {Machine review of arXiv:2607.21259}
}
abstract
We study a half-interval distribution problem for polynomial residues modulo an odd prime $p$: how often the fractional part of $\varphi(x)/p$ lies in the upper half of the unit interval as $x$ ranges over $1\leq x< p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\left\{1\leq x< p/2:\left\{{\varphi(x)}/{p}\right\}>\frac12\right\} =\frac{p}{4}+O_\varphi(\sqrt p\log^2 p). $ We then show that the error term can be improved to $O_\varphi(\sqrt p\log p)$ for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials $\varphi(x)=x^m$, we further obtain the bound $O_m(\sqrt p\log\log p)$ under the Generalized Riemann Hypothesis. Finally, in the case $m=2$, we prove an unconditional matching lower bound, showing that the factor $\log\log p$ is best possible in this setting.
Reference graph
Works this paper leans on
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