Squarefree numbers in short intervals: explicit and formalized

Mayank Pandey

REVIEW 3 major objections 4 minor 300 references

Squarefree numbers in short intervals: explicit and formalized

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper makes explicit and machine-checks a power-saving error bound for squarefree numbers in short intervals: for admissible ε and enormous X, with H = X^(1/5 − 2/90935 + ε), the squarefree count in [X, X+H] is within (10^450/ε) H…

desk verdict The formalization is the real claim and the real risk; the printed note cannot carry the theorem alone, but the approach is honest and deserves a serious referee. read the letter →

arxiv 2608.06682 v1 pith:SVKWNFSN submitted 2026-08-07 math.NT

Mayank Pandey This is my paper · ORCID

classification math.NT MSC 11N2511N3711L07

keywords squarefreenumbersshortintervalsexplicitestimatesvanderCorputdifferencingbracketpolynomialsformalverificationpower-savingerror

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 gives a fully explicit version of the author's earlier result on squarefree numbers in short intervals: for any $0 < \varepsilon \le 1/90935$ and $X \ge \exp(10^{27}/\varepsilon^2)$, with $H = X^{1/5 - 2/90935 + \varepsilon}$, the number of squarefree integers in $[X, X+H]$ is within $(10^{450}/\varepsilon)\,H\,X^{-\varepsilon/10^{25}}$ of the expected value $(6/\pi^2)H$. The interest is that both the exponent and every constant are pinned down, and the proof is recorded in a machine-checked formalization. The note explains the main new step: the previously nonexplicit range, which came from quantitative equidistribution of nilsequences, is handled here by elementary van der Corput differencing applied to a bracket expression, together with a union bound over finitely many integer carry shifts. If correct, this removes the last ineffective ingredient from the estimate.

What carries the argument

The central object is $N(H,D)$, the count of $d\in[D,2D]$ for which the fractional part of $X/d^2$ lands within $H/D^2$ of an integer. The argument bounds $N(H,D)$ by splitting into ranges and applying three propositions; the crucial new tool is the explicit finite-difference identity (9) for the bracket expression $g(r)$, which lets the author reduce nonconcentration of $g$ to a union bound over $O(1)$ integer carry shifts $\rho_0,\ldots,\rho_3$. Combined with van der Corput differencing and an elementary estimate for lattice points on a convex curve, this gives Proposition 3.1, the critical-range estimate that makes the exponent explicit.

What would settle it

Since the stated $X$ is astronomically large, a direct numerical search cannot reach the theorem's range; the concrete check is to verify Proposition 3.1's bound for $N(H,D)$ on the critical scale, for example by taking a small but admissible $X$ and admissible $\varepsilon$, choosing $D$ in the critical range (7), and searching for $d \in [D,2D]$ with $\|X/d^2\| \le H/D^2$. A single $d$ exceeding the claimed bound would invalidate the key step, as would a flaw found in the formal derivation of Proposition 3.1.

Watch

Extended reading notes

Core claim

Theorem 1.1 states that the deviation of the squarefree count in an interval of length $H$ from the expected $(6/\pi^2)H$ is at most an explicit power-saving multiple of $H X^{-\varepsilon/10^{25}}$. The proof reduces the count to bounding $N(H,D) = \#\{d\in[D,2D] : \|X/d^2\| \le H/D^2\}$ across all relevant scales of $D$. Three propositions cover the easy ranges, and the critical range, previously handled by a nonexplicit equidistribution theorem, is covered by Proposition 3.1, an upper bound obtained by finite differencing of the bracket expression $g(r) = f_3(r) + f_1(r)\{f_2(r)\}$. The finite-difference identity (9) shows that after differencing three times, the bracket expression reduces to a sum of monomials plus a small error, with the carry information encoded in finitely many bounded integers; the paper splits over those possibilities by a union bound instead of proving equidistribution. The assembled result is an explicit bound for $N(H,D)$ in the critical range, which yields Theorem 1.1.

Load-bearing premise

The whole explicit bound depends on Proposition 3.1, a bound for the critical range that the note states without proof and leaves to the accompanying formal proof files; if that proposition is wrong, Theorem 1.1 collapses.

Editorial extensions

If this is right

If Theorem 1.1 is correct, the error term $H X^{-\varepsilon/10^{25}}$ gives a fully explicit power saving for every admissible $\varepsilon$, not just an asymptotic one.
Every constant in the estimate is computable, so the bound can be used as a black box in other arguments without worrying about ineffective constants.
The critical range no longer depends on a nonexplicit equidistribution theorem; the proof uses only elementary differencing and a union bound over carry shifts.
The machine-checked formalization means the statement of Theorem 1.1 has been verified mechanically, assuming the correctness of the formal proof system itself.
The explicit exponent $1/5 - 2/90935$ is the bottleneck produced by Proposition 3.1; any improvement in that proposition would directly improve the exponent in the theorem.

Reading between the lines

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

The union-bound over integer carry shifts in identity (9) is a general recipe: any problem where a nilsequence reduces to a bracket polynomial with integer-valued floors could be made explicit in the same way, without a full equidistribution theorem.
Because Proposition 3.1 is the only unproved ingredient in the expository text, the formal proof files, not the note itself, are the real authority for the critical step; a reader who cannot inspect them must take that proposition on faith.
The exponent $2/90935$ is not likely to be optimal; the author explicitly trades some sharpness for simplicity of exposition, so tightening the choices in Proposition 3.1 would plausibly yield a larger power saving.
The explicit nature of the bound invites a computational check on intermediate scales: even though the full $X$ range is out of reach, the inequality for $N(H,D)$ in the critical scale could be tested numerically to probe whether the shape of Proposition 3.1 is plausible.

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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

First, the actual news: Pandey gets an explicit power-saving bound for squarefree numbers in short intervals by replacing the non-explicit Green–Tao input with elementary van der Corput differencing, and he claims a Lean 4 formalization of the whole thing. The exponent 2/90935 and the enormous X lower bound are far from practical, but that is not the point. An explicit, machine-checked theorem of this classical type is a genuine milestone, and the candid account of AI-assisted formalization is useful for people thinking about how to do serious verification work. The paper is honest about what is formalized and does not oversell the mathematics.

The soft spots are real, though they are mostly about verifiability, not about the mathematical spine. Proposition 3.1, the critical-range bound, is stated only asymptotically, with an e, a U^O(1), and unspecified implied constants. The text alone cannot produce the 10^450 constants in Theorem 1.1. Remark 1 explicitly concedes that the printed proposition is a weakening of what is actually proved in Lean: the formalized result bounds N_a(H,D) and the relevant ranges depend on a. So the printed lemmas do not fully match the proof, and the repository is not pinned to a commit, with no build or verification instructions. A reader who cannot or will not dig through the Lean code has no way to check the key step. That is a serious accessibility gap for a paper whose main claim to verification is the formalization.

I do not think the paper is circular or fudging. The argument is conditional on the correctness of the formalization, and the authors are explicit about the gap between the note and the code. The differencing identity (9) and the union-bound over carry shifts are plausible and consistent with the earlier framework. There is no fitting or definitional sleight of hand here. So the central claim is not obviously wrong; it is simply unverifiable from the PDF.

For peer review, this should be engaged with, not desk-rejected. A serious referee could check the repository, pin the commit, and run the Lean code. The journal should require that the repo be pinned and that either Proposition 3.1 be stated explicitly in its true formalized form or the full proof be made available in a documented supplementary file. Without those, the printed note is not self-contained. I would cite this only after the repo is pinned and verified, not before. Reading group: maybe, as a data point about automated formalization, but not for the number theory itself.

Referee Report

3 major / 4 minor

Summary. The manuscript announces an explicit, Lean-formalized power-saving estimate for the number of squarefree numbers in short intervals. Theorem 1.1 states that, for 0 < ε ≤ 1/90935, X ≥ exp(10^27/ε^2), and H = X^{1/5 - 2/90935 + ε}, the deviation of the squarefree count in [X, X+H] from (6/π^2)H is at most (10^450/ε) H X^{-ε/10^25}. The argument follows the author's earlier work [Pan24], replacing the Green-Tao nilsequence input in the critical range by elementary van der Corput differencing on a bracket expression, and states the necessary critical-range bound as Proposition 3.1. The paper explicitly acknowledges in Remark 1 that the printed propositions are weakened versions of the statements actually formalized in the linked Lean repository.

Significance. If the formalized proof is correct, this is a substantial result: it gives the first fully explicit, machine-checked power saving for squarefree numbers in short intervals, with a printed constant of order 10^450. The decision to avoid Fourier analysis by using elementary Swinnerton-Dyer bounds is a useful simplification that may make the argument easier to formalize and verify. The existence of a machine-checked proof is a genuine strength, but it is only accessible if the repository is pinned and fully inspectable. As submitted, the manuscript does not provide enough information to verify the central claim from the printed text alone, which substantially limits the significance of the announcement.

major comments (3)

[§3, Proposition 3.1] Proposition 3.1 is the critical-range input for Theorem 1.1, but as printed it cannot support the theorem. It asserts N(H,D) ≪ H (1/U + U^{O(1)}(...)) with an unspecified constant in the O(1) exponent and an implied constant in the ≪ notation. Since Theorem 1.1 is an explicit bound with constants such as 10^450 and exponent -ε/10^25, the reader cannot pass from (10) to (1). The text also gives no proof of Proposition 3.1: it only cites Lean identifiers. In particular, the derivation from the differencing identity (9) is not shown, including how the union bound over the integer carry shifts ρ_i yields the displayed exponents in (10). I request that the explicit form of Proposition 3.1 with all constants be stated and proved in the text or in an appendix.
[Remark 1] Remark 1 concedes that the printed statements are weakened proxies for the formalized results. Specifically, the formalized object is N_a(H,D), and the applicability of Squarefree.prop_5_1 versus Squarefree.prop_7_3 depends on the scale of a as well as on Δ. The manuscript does not state the explicit bound for N_a(H,D) that is actually proved, nor does it explain how Theorem 1.1 follows from bounds on N_a rather than on N(H,D). This leaves a gap in the logical chain from the displayed propositions to the theorem.
[§2, Propositions 2.1–2.3] The preparatory propositions are also asymptotic statements with omitted constants and, in Proposition 3.1, an unspecified U^{O(1)}. Proposition 2.2 gives C=10^20 but later uses require a precise relation among U, Δ, G, H, and ε. The manuscript does not show the optimization step that selects U and Δ so that the combination of Propositions 2.3 and 3.1 yields a bound of the form H X^{-ε/10^25}. Without this calculation, the claimed numerical value 10^450 in Theorem 1.1 cannot be checked from the text.

minor comments (4)

[Title] The title as printed contains typographical spacing errors: 'SHOR T INTER V ALS' should be 'SHORT INTERVALS'.
[Repository information] The GitHub repository is referenced by URL but is not pinned to a commit, and no build instructions or specific Lean version are given. For a formalization to be verifiable, the exact commit and environment are needed.
[Equation (9)] The summation notation in (9) is difficult to parse: '3X i≤3 j<ks.t.{1,2,3}={i,j,k}' appears to be a mangled summation; please use standard sum notation with explicit index sets.
[Page 2] The sentence 'This role was played in the original paper by [Pan24, Proposition 2.2], which is a bit stronger then 2.1' contains a typo: 'then' should be 'than'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is a new explicit estimate whose proof depends on independent formalized bounds, not on definitions or fits that encode the conclusion.

full rationale

The paper's advertised result is an explicit, formalized squarefree-counting bound. I looked for each circular pattern. The argument uses [Pan24] for setup (layout, reduction to N(H,D), elementary differencing) and [SD74] for an elementary lattice-point estimate; those are prior independent results and are not used to force the final constant. The critical new input is Proposition 3.1, an upper bound for N(H,D) obtained by van der Corput differencing on the bracket expression (8)-(9), followed by a union bound over integer carry shifts. Nothing in that argument is defined in terms of the count deviations in Theorem 1.1; it is a genuine estimate on a different counting quantity. The proposition is stated without proof in the note and referred to the Lean repository, and Remark 1 concedes that the printed statements are weakened variants of the formalized N_a(H,D) bounds. Those are completeness/reproducibility issues (the repository is not pinned and the printed proposition is not self-contained), not instances of conclusion-from-input circularity. There is no fitted parameter later called a prediction, no uniqueness theorem imported from the authors, and no ansatz disguised as an external citation. Even if Proposition 3.1 is wrong or the printed derivation is too weak to assemble without the repository, the failure mode would be an unsupported or missing proof, not circularity. Under the hard rules, an unproved lemma is a correctness risk; it does not make the derivation equivalent to its inputs. Accordingly, score 0.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central proof relies on standard elementary estimates and on the setup from [Pan24]; no new postulates or entities are introduced.

assumptions (2)

domain assumption Swinnerton-Dyer's bound on the number of lattice points on convex curves [SD74]
Used in Proposition 2.2 and in the formalization in place of [Pan24, Proposition 3.3] to avoid Fourier analysis.
domain assumption The reductions and framework of [Pan24]
The paper's setup, including the reduction to N(H,D) and the finer quantity N_a(H,D), is imported from [Pan24] and not reproved in this note.

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Cite this review

Pith. "Pith review of Squarefree numbers in short intervals: explicit and formalized." pith.science (2026). https://pith.science/paper/SVKWNFSN

@misc{pith2026260806682,
  author       = {Pith},
  title        = {Pith review of: Squarefree numbers in short intervals: explicit and formalized},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SVKWNFSN}},
  note         = {Machine review of arXiv:2608.06682}
}

abstract

We make explicit and formalize a result of the author on squarefree numbers in short intervals, showing that for $0 < \varepsilon\le 1/90935 $, $X\ge \exp(10^{27}/\varepsilon^2)$, $H = X^{1/5 - 2/90935 + \varepsilon}$, we have that \[ \biggl|\sum_{X\le n\le X + H } \mu(n)^2 - \frac{6}{\pi^2}H\biggr| \le \frac{10^{450}}{\varepsilon} H X^{-\varepsilon/10^{25}}. \] This article gives an account of what went into making the exponent explicit. The Github repository linked contains the formalization in Lean 4 as well as an account of what went into the largely automated formalization.

Discussion (0). Continue with ORCID to comment.

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Oleksiy Klurman and Youness Lamzouri and Marc Munsch , display-string =. Sign changes of short character sums and real zeros of Fekete polynomials , url =
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Raghuram , display-string =

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Sheng-Yang Kevin Ho , display-string =. The Rational Torsion Subgroup of J_0(
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Edgar Assing , display-string =. An observation concerning highly ramified ε -factors , url =
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Valentin Blomer and Subhajit Jana and Paul D. Nelson , display-string =. Local integral transforms and global spectral decomposition , url =
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Alexandre de Faveri and Zvi Shem-Tov , display-string =. Non-escape of mass for arithmetic quantum limits on hyperbolic 4 -manifolds , url =
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William Craig and Jan-Willem van Ittersum and Ken Ono , display-string =. Integer partitions detect the primes , url =
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Chen and Akshay Venkatesh , display-string =

Eric Y. Chen and Akshay Venkatesh , display-string =. Some Singular Examples of Relative Langlands Duality , url =
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Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality , url =

Zhicheng Wang , display-string =. Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality , url =
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Marios Voskou , display-string =. Refined Counting of Geodesic Segments in the Hyperbolic Plane , url =
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Owen Barrett and Zoë X. Batterman and Aditya Jambhale and Steven J. Miller and Akash L. Narayanan and Kishan Sharma and Chris Yao , display-string =. A Random Matrix Model for a Family of Cusp Forms , url =
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Anke Pohl and YoungJu Choie and Roelof Bruggeman , display-string =. Period functions for vector-valued Maass cusp forms of real weight, with an application to Jacobi Maass cusp forms , url =
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Andre Reznikov , display-string =. Mass of the Frobenius functional on small balls , url =
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Relative Langlands Duality , url =

David Ben-Zvi and Yiannis Sakellaridis and Akshay Venkatesh , display-string =. Relative Langlands Duality , url =
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Local Intertwining Relations and Co-tempered A -packets of Classical Groups , url =

Hiraku Atobe and Wee Teck Gan and Atsushi Ichino and Tasho Kaletha and Alberto Mínguez and Sug Woo Shin , display-string =. Local Intertwining Relations and Co-tempered A -packets of Classical Groups , url =
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Young , display-string =

Yueke Hu and Ian Petrow and Matthew P. Young , display-string =. A generalized PGL(2) Petersson/Bruggeman/Kuznetsov formula for analytic applications , url =
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Shparlinski , display-string =

Pär Kurlberg and Alina Ostafe and Zeev Rudnick and Igor E. Shparlinski , display-string =. On quantum ergodicity for higher dimensional cat maps , url =
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Schubert cells and Whittaker functionals for

Doyon Kim , display-string =. Schubert cells and Whittaker functionals for
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Introduction to Shtukas and their moduli , url =

Zhiwei Yun , display-string =. Introduction to Shtukas and their moduli , url =
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Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions , url =

Philippe Michel and Dinakar Ramakrishnan and Liyang Yang , display-string =. Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions , url =
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Formes modulaires modulo 2 : L'ordre de nilpotence des opérateurs de Hecke (version développée) , url =

Jean-Louis Nicolas , display-string =. Formes modulaires modulo 2 : L'ordre de nilpotence des opérateurs de Hecke (version développée) , url =
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Projective smooth representations in natural characteristic , url =

Amit Ophir and Claus Sorensen , display-string =. Projective smooth representations in natural characteristic , url =
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Eisenstein Series on Metaplectic Covers and Multiple Dirichlet Series , url =

Yanze Chen , display-string =. Eisenstein Series on Metaplectic Covers and Multiple Dirichlet Series , url =
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Mean Value for Random Ideal Lattices , url =

Nihar Gargava and Maryna Viazovska , display-string =. Mean Value for Random Ideal Lattices , url =
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A characterization of Jacobi sums , url =

Andrew Snowden , display-string =. A characterization of Jacobi sums , url =
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Real quadratic base changes for

Tristan Ricoul , display-string =. Real quadratic base changes for
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V. K. Dobrev , display-string =. Langlands Duality and Invariant Differential Operators , url =
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Construction of eigenvarieties , url =

James Newton , display-string =. Construction of eigenvarieties , url =
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Valentin Blomer and Igor E. Shparlinski and Morten S. Risager , display-string =. Triple sums of Kloosterman sums and the discrepancy of modular inverses , url =
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Hecke algebras and local Langlands correspondence for non-singular depth-zero representations , url =

Maarten Solleveld and Yujie Xu , display-string =. Hecke algebras and local Langlands correspondence for non-singular depth-zero representations , url =
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Spectral Reciprocity and Hybrid Subconvexity Bound for triple product L -functions , url =

Xinchen Miao , display-string =. Spectral Reciprocity and Hybrid Subconvexity Bound for triple product L -functions , url =
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Xinyu Guan and Li Lyna Zhang and Yifei Liu and Ning Shang and Youran Sun and Yi Zhu and Fan Yang and Mao Yang , display-string =. rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking , url =
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Pith tools

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Squarefree numbers in short intervals , url =

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Rydin Myerson and Pankaj Vishe , display-string =

Junxian Li and Simon L. Rydin Myerson and Pankaj Vishe , display-string =. A two-dimensional delta symbol method and its application to pairs of quadratic forms , url =

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