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On the Gauss circle problem over smooth numbers

This paper provides asymptotic evaluations for the sum of the number of representations of $y$ -smooth numbers as sums of two squares for specific ranges of $x$ and $y$ .

Original authors: Peng Gao

Published 2026-04-28

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Peng Gao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Concept: The "Smooth" Circle Problem

Imagine you have a giant sheet of graph paper. You draw a large circle on it, and you want to count how many of the "intersections" (the points where the grid lines cross) fall inside that circle.

In mathematics, this is the Gauss Circle Problem. The number of points inside the circle is roughly equal to the area of the circle ( $\pi r^2$ ), but because the grid is made of discrete points, there is always a little bit of "error"—some points are just barely inside, and some are just barely outside. Mathematicians spend their lives trying to figure out exactly how large that "error" can be.

But this paper adds a twist.

Instead of counting every intersection, the author, Peng Gao, is only interested in a special "VIP club" of points. These are called "Smooth Numbers."

The Metaphor: The VIP Party

Imagine you are hosting a massive gala (the Circle) inside a ballroom. You invite everyone in the city (all the integers).

However, you have a very strict security guard at the door. He only lets in guests who are "Smooth." In math, a number is "smooth" if all its "building blocks" (its prime factors) are small.

A "Rough" Number is like a guest who arrives in a massive, heavy armored truck (a huge prime number like 997). They are hard to manage and "clunky."
A "Smooth" Number is like a guest who arrives in a fleet of tiny, nimble bicycles (small prime factors like 2, 3, and 5). They are easy to organize and fit into small spaces.

The Question: If we only count the "smooth" guests inside our circle, how many will there be? And how does that number change as the circle gets bigger?

What the Paper Actually Does

The paper provides a mathematical "formula" (an asymptotic evaluation) to predict the number of these smooth guests.

1. Finding the "Saddle Point" (The Sweet Spot)

To solve this, Gao uses a technique called the Saddle Point Method.
Imagine you are trying to balance a marble on a curved surface. There is one specific spot where the marble sits perfectly still. In this math problem, the "Saddle Point" is the perfect mathematical "sweet spot" that allows the author to calculate the density of these smooth numbers without getting lost in the infinite complexity of the grid.

2. The Dickman Function (The Predictor)

The paper mentions the Dickman Function. Think of this as a "probability weather report." It tells you: "As your circle gets larger, what is the likelihood that a random guest will be 'smooth' enough to pass security?" Gao uses this to refine his prediction.

3. Expanding the Range (The Breakthrough)

Previous mathematicians had already studied this, but they could only predict the number of smooth guests if the circle was a certain size or if the "security rules" were very specific.

Gao’s contribution is like upgrading the radar. He proved that his formula works for a much wider variety of scenarios—even when the circle is massive or when the "smoothness" requirement changes drastically.

Summary for a Non-Mathematician

If the original Gauss Circle Problem is about counting all the dots in a circle, Peng Gao’s paper is about counting only the "easy-to-handle" dots (the smooth numbers).

He has created a highly accurate mathematical "map" that tells us exactly how many of these special dots to expect, even in extreme conditions where previous maps failed. He did this by finding the "perfect balance point" (the saddle point) in a sea of complex prime numbers.

This paper, titled "On the Gauss Circle Problem over Smooth Numbers" by Peng Gao, investigates the distribution of the number of ways to represent integers as sums of two squares, specifically when those integers are "smooth" (or "friable").

Below is a detailed technical summary of the research.

1. The Problem: Gauss Circle Problem over Smooth Numbers

The classical Gauss Circle Problem asks for the number of integer lattice points $(m, n)$ inside a circle of radius $\sqrt{x}$ , which is equivalent to evaluating the sum:
$\sum_{n \le x} r(n)$
where $r(n)$ is the number of representations of $n$ as a sum of two squares. Gauss proved this is $\pi x + O(x^{1/2})$ , and modern research focuses on minimizing the error term.

The author extends this problem to $y$ -smooth numbers. An integer is $y$ -smooth if its largest prime factor $P(n)$ satisfies $P(n) \le y$ . The central object of study is the weighted sum:
$\Psi_G(x, y) := \sum_{\substack{n \le x \\ P(n) \le y}} r(n)$
The goal is to find an asymptotic formula for $\Psi_G(x, y)$ that remains valid for a much wider range of the parameter $u = \frac{\log x}{\log y}$ than previously established in existing literature.

2. Methodology: The Saddle Point Method

The paper employs the Saddle Point Method (also known as the method of steepest descent), a sophisticated technique in analytic number theory. The methodology follows these steps:

Dirichlet Series and Euler Products: The author defines a generating function $H(s, G; y) = \sum_{P(n) \le y} \frac{r(n)/4}{n^s}$ . Because $r(n)/4$ is a multiplicative function, this can be expressed as an Euler product involving the Riemann zeta function $\zeta(s)$ and the Dirichlet $L$ -function $L(s, \chi_4)$ (where $\chi_4$ is the primitive character modulo 4).
Rankin’s Trick: To bound the sum, the author uses Rankin's trick to relate the sum to the generating function evaluated at a real parameter $\sigma > 0$ .
Saddle Point Selection: The author identifies a unique "saddle point" $\alpha_G(x, y)$ , which is the real solution to the equation $\frac{d}{ds} \log H(s, G; y) + \log x = 0$ . This point is chosen to minimize the integrand in the Perron formula.
Perron’s Formula: The sum is expressed as a complex contour integral. The author shifts the contour to the line $\Re(s) = \alpha_G$ and uses estimates for the $L$ -functions and the distribution of primes to evaluate the integral.

3. Key Contributions and Results

The paper provides two primary mathematical results:

Theorem 1.1: The Asymptotic Formula

The author establishes a new asymptotic formula for $\Psi_G(x, y)$ for the range $(\log \log y)^2 \le u \le \frac{y^{1/(2+\epsilon_0)}}{\log y}$ :
$\Psi_G(x, y) = \frac{4x^{\alpha_G} H(\alpha_G, G; y)}{\alpha_G \sqrt{2\pi \phi_2(\alpha_G, G; y)}} \left( 1 + O\left(\frac{1}{u}\right) \right)$
This result is significant because it extends the range of $u$ (the ratio of the logs of $x$ and $y$ ) significantly beyond the previous bounds set by A. Goswami.

Theorem 1.2: Explicit Behavior of the Saddle Point

The author provides explicit asymptotic behaviors for the saddle point $\alpha_G$ and the second derivative of the log-generating function $\phi_2$ . Crucially, the author shows that the main term is consistent with the Dickman function $\rho(u)$ , which governs the density of smooth numbers. This links the Gauss circle problem directly to the classical theory of smooth numbers.

4. Technical Significance

Extension of Range: The primary achievement is the widening of the valid range for $u$ . This allows for a more granular understanding of how the "circle problem" behaves as the smoothness constraint $y$ changes relative to $x$ .
Analytic Sophistication: The proof requires delicate handling of the zero-free regions of the Dedekind zeta function for the Gaussian number field $\mathbb{Q}(i)$ . The author successfully manages the error terms arising from the $L$ -function $L(s, \chi_4)$ and the prime sums.
Consistency: The results bridge the gap between the distribution of smooth numbers (the $\Psi(x, y)$ problem) and the arithmetic properties of sums of two squares, showing that the "smoothness" of the integers dominates the asymptotic behavior of the sum.