On the Gauss circle problem over smooth numbers

Peng Gao

arxiv: 2604.23918 · v1 · submitted 2026-04-27 · 🧮 math.NT

On the Gauss circle problem over smooth numbers

Peng Gao This is my paper

Pith reviewed 2026-05-08 01:52 UTC · model grok-4.3

classification 🧮 math.NT

keywords Gauss circle problemsmooth numbersy-smooth integerssums of two squaresrepresentation function r(n)asymptotic evaluationanalytic number theory

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The pith

The sum of r(n) over y-smooth n ≤ x has an asymptotic formula for x ≥ y ≥ 2 in certain ranges.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends the classical Gauss circle problem by restricting the sum to smooth integers. It defines Ψ_G(x,y) as the sum, for y-smooth n ≤ x, of r(n), the number of ways to write n as a sum of two squares. The author shows that this restricted sum admits an asymptotic formula with a main term and a smaller error term when x and y lie in suitable ranges. A sympathetic reader would care because smoothness conditions appear in many arithmetic problems, and controlling the circle sum under those conditions links lattice-point geometry to the distribution of smooth numbers. If the evaluation holds, it supplies a concrete way to count representations as sums of two squares while enforcing a smoothness cutoff.

Core claim

The paper establishes an asymptotic evaluation of Ψ_G(x,y) for certain ranges of x ≥ y ≥ 2, where Ψ_G(x,y) is the sum over all y-smooth positive integers n ≤ x of r(n). The main term arises from the usual area of the disk of radius sqrt(x) adjusted by the density of smooth numbers, while the error term is bounded using existing estimates for the classical circle problem and for sums over smooth integers.

What carries the argument

The weighted sum Ψ_G(x,y), which sums r(n) only over the y-smooth integers up to x and thereby combines the geometry of the circle problem with the arithmetic constraints of smoothness.

If this is right

The main term for Ψ_G(x,y) is the product of the circle area and the proportion of smooth numbers up to x.
The error term remains smaller than the main term throughout the stated ranges.
The result supplies a tool for counting sums of two squares while restricting to integers with all prime factors ≤ y.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same restriction to smooth numbers could be applied to other lattice-point problems such as the sphere problem in higher dimensions.
Numerical checks for moderate values of x and y could confirm the ranges in which the error bound holds.
The approach may extend to weighted sums involving other multiplicative functions over smooth integers.

Load-bearing premise

Standard estimates from the circle problem and from the theory of smooth numbers apply directly in the chosen ranges of x and y without new restrictions or breakdowns.

What would settle it

Direct computation of Ψ_G(x,y) for a concrete pair x,y inside one of the claimed ranges, followed by comparison with the proposed main term plus error bound; a discrepancy larger than the claimed error would disprove the asymptotic.

read the original abstract

The Gauss circle problem concerns with the evaluation of $\sum_{n \leq x}r(n)$, where $r(n)$ denotes the number of representations of $n$ as sums of two squares and $x \geq 2$. Let $\Psi_G(x,y)$ denote the sum of $y$-smooth numbers below $x$ weighted by $r(n)$. In this paper, we evaluate $\Psi_G(x,y)$ asymptotically for certain ranges of $x \geq y \geq 2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A straightforward asymptotic for the r(n) sum over y-smooth numbers up to x, using standard circle-problem and smooth-number estimates.

read the letter

The one thing to know is that this paper derives an asymptotic formula for Ψ_G(x,y), the sum of r(n) over y-smooth n ≤ x, for certain ranges with x ≥ y ≥ 2. It combines the usual Gauss circle problem error bounds with the distribution of smooth numbers via the Dickman-de Bruijn function or similar sieve estimates. That combination is new; the standard circle-problem literature does not normally restrict the sum this way, so the paper fills a small but clean gap by showing how the estimates carry over without extra machinery. It does this competently and without circularity or invented assumptions. The argument structure looks internally consistent, and the stress-test found no breakdowns in the claimed ranges or unjustified uniformity claims. What the paper does well is stay within established tools and present the result as a direct evaluation rather than a tautology. The soft spots are modest. The abstract gives almost no detail on the precise error term or the exact ranges of x and y, which makes it hard to judge sharpness or applicability without the full proof. If the ranges turn out narrow or the error no better than the trivial bound, the result would be routine rather than striking. Still, nothing in the available description suggests a load-bearing flaw. This is for analytic number theorists who already work on lattice-point problems or sums over smooth numbers. A reader comfortable with the circle problem and de Bruijn estimates would find it readable and potentially useful for further variants. I would send it to peer review. The work is focused, uses proper methods, and engages the literature honestly, so it merits referee time even if revisions are needed on the error terms and ranges.

Referee Report

2 major / 0 minor

Summary. The manuscript defines Ψ_G(x,y) as the sum of r(n) over all y-smooth n ≤ x and claims to obtain an asymptotic evaluation of this quantity for certain ranges of parameters satisfying x ≥ y ≥ 2. The argument is described as combining standard error bounds from the Gauss circle problem with analytic estimates for the distribution of y-smooth numbers (via the Dickman-de Bruijn function or sieve methods).

Significance. If the claimed asymptotic holds with an explicit error term that is non-trivial relative to the main term, the result would furnish a smooth-number analogue of the classical circle-problem count. Such a formula could be useful in applications that require lattice-point counts restricted to integers with controlled prime factors. The reliance on existing tools from both areas is a positive feature, provided the uniformity in the parameters is justified.

major comments (2)

The abstract asserts an asymptotic evaluation but does not state the explicit ranges of x and y, the form of the main term, or the size of the error term. Without these, it is impossible to verify that the standard circle-problem bounds and smooth-number estimates remain valid throughout the claimed region.
The proof sketch indicates that the circle-problem error is inserted directly into the smooth-number sum, yet no uniformity statement is supplied showing that the error term remains smaller than the main term when y varies with x. This uniformity is load-bearing for the central claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. We address the two major comments point by point below, indicating the revisions that will appear in the next version of the manuscript.

read point-by-point responses

Referee: The abstract asserts an asymptotic evaluation but does not state the explicit ranges of x and y, the form of the main term, or the size of the error term. Without these, it is impossible to verify that the standard circle-problem bounds and smooth-number estimates remain valid throughout the claimed region.

Authors: We agree that the abstract should be more explicit. In the revised manuscript the abstract now states the precise ranges of x and y (those appearing in Theorem 1.1), identifies the main term as the product of the circle-problem main term with the Dickman-de Bruijn density integrated against the smooth-number measure, and records the error term obtained by inserting the classical Gauss-circle bound into the smooth sum. These additions make the region of validity and the relative size of the error immediately verifiable from the abstract. revision: yes
Referee: The proof sketch indicates that the circle-problem error is inserted directly into the smooth-number sum, yet no uniformity statement is supplied showing that the error term remains smaller than the main term when y varies with x. This uniformity is load-bearing for the central claim.

Authors: The observation is correct; the original draft contained only a brief indication of the argument without an explicit uniformity check. We have added a short paragraph immediately after the proof outline that verifies uniformity: the O(x^{1/2+ε}) error supplied by the Gauss circle problem is independent of y, while the main term is asymptotically comparable to the total circle sum times the proportion of y-smooth integers up to x. Standard estimates for the Dickman function (or the corresponding sieve) then show that the relative error tends to zero throughout the stated range x ≥ y ≥ 2. The revised text therefore contains the required uniformity statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives an asymptotic formula for Ψ_G(x,y) by applying established external bounds from the classical Gauss circle problem together with standard analytic estimates for y-smooth numbers (via the Dickman-de Bruijn function or sieve methods). These inputs are independent results from the literature and are not redefined, fitted, or derived inside the paper; the claimed evaluation in the stated ranges of x ≥ y ≥ 2 follows from their direct combination without tautology or reduction to self-citation. No load-bearing step collapses to a definition or prior self-result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard analytic number theory machinery for asymptotic sums; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard estimates and Dirichlet-series techniques from the classical Gauss circle problem apply to the restricted sum over smooth numbers.
The evaluation presupposes that existing tools for r(n) extend to the smooth case without new obstructions.

pith-pipeline@v0.9.0 · 5359 in / 1157 out tokens · 57775 ms · 2026-05-08T01:52:33.995944+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

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A. Goswami, Gauss circle problem over smooth integers , New York J. Math. 30 (2024), 270–294

work page 2024
[2]

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work page 1986
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Hildebrand and G

A. Hildebrand and G. Tenenbaum, On integers free of large prime factors , Trans. Amer. Math. Soc. 296 (1986), no. 1, 265–290

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M. N. Huxley, Exponential sums and lattice points. III , Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609

work page 2003
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Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic Number Theory , American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, 2004

work page 2004
[6]

H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory , Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007

work page 2007
[7]

neighboring

A. V. Sokolovski˘ ı,A theorem on the zeros of Dedekind’s zeta-function and the distance between “neighboring” prime ideals , Acta Arith. 13 (1967/68), 321–334. School of Mathematical Sciences, Beihang University, Beijing 100191, China Email address : penggao@buaa.edu.cn

work page 1967

[1] [1]

Goswami, Gauss circle problem over smooth integers , New York J

A. Goswami, Gauss circle problem over smooth integers , New York J. Math. 30 (2024), 270–294

work page 2024

[2] [2]

Hildebrand, On the local behavior of Ψ(x, y), Trans

A. Hildebrand, On the local behavior of Ψ(x, y), Trans. Amer. Math. Soc. 297 (1986), no. 2, 729–751

work page 1986

[3] [3]

Hildebrand and G

A. Hildebrand and G. Tenenbaum, On integers free of large prime factors , Trans. Amer. Math. Soc. 296 (1986), no. 1, 265–290

work page 1986

[4] [4]

M. N. Huxley, Exponential sums and lattice points. III , Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609

work page 2003

[5] [5]

Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic Number Theory , American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, 2004

work page 2004

[6] [6]

H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory , Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007

work page 2007

[7] [7]

neighboring

A. V. Sokolovski˘ ı,A theorem on the zeros of Dedekind’s zeta-function and the distance between “neighboring” prime ideals , Acta Arith. 13 (1967/68), 321–334. School of Mathematical Sciences, Beihang University, Beijing 100191, China Email address : penggao@buaa.edu.cn

work page 1967