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A link between error terms when counting smooth and rough numbers

This paper establishes a mathematical relationship between the error terms in the counting functions of smooth and rough numbers, using this link to derive an explicit upper bound for the error in de Bruijn's approximation for smooth numbers based on existing bounds for rough numbers.

Original authors: Andreas Weingartner

Published 2026-04-27

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Andreas Weingartner

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 Tale of the Smooth and the Rough: A Mathematical Bridge

Imagine you are sorting a massive pile of numbered tiles. In this world, numbers have "personalities" based on their building blocks (their prime factors).

There are two main groups of numbers:

The "Smooth" Numbers: These are the "easy-going" numbers. They are made up only of small, gentle prime factors (like 2, 3, and 5). They don't have any "big, scary" prime factors.
The "Rough" Numbers: These are the "tough" numbers. They are made up only of large, intimidating prime factors. They avoid the small, common numbers entirely.

For decades, mathematicians have been trying to count exactly how many of these numbers exist up to a certain point. Because these numbers are scattered in complex patterns, mathematicians use "approximations"—basically, high-level educated guesses—to estimate the totals.

The Problem: The "Error Gap"
Even the best guesses aren't perfect. There is always a tiny gap between the actual count and the estimated count. This gap is called the Error Term.

For a long time, mathematicians studied the errors for "Smooth" numbers and "Rough" numbers as if they were two different species living on different planets. They knew how to estimate the errors for each, but they didn't realize they were looking at two sides of the same coin.

What This Paper Does: Building the Bridge

Andreas Weingartner’s paper is essentially a mathematical bridge. He proves that the error in counting Smooth numbers and the error in counting Rough numbers are mathematically "linked."

The Analogy: The Mirror and the Shadow

Imagine you are standing in a room with a bright light.

The Smooth numbers are like the Object itself.
The Rough numbers are like the Shadow cast by that object.

If the object moves slightly (an error in the Smooth count), the shadow must also move in a predictable way (an error in the Rough count). If you know exactly how the shadow is wobbling, you can use math to figure out exactly how the object is vibrating.

Weingartner provides the "formula" for this relationship. He shows that if you have a way to measure the "wobble" (the error) in the Rough numbers, you can plug it into his equations to get a precise measurement of the "wobble" in the Smooth numbers.

Why Does This Matter? (The "So What?")

In the paper, the author uses this new bridge to do something very practical: He makes a better guess.

He takes an existing, highly accurate way to measure the error in "Rough" numbers (a method developed by a mathematician named Fan) and "walks across the bridge" to create a new, explicit upper bound for the error in "Smooth" numbers.

In plain English: He took a high-quality ruler used for measuring shadows and used it to build a much more accurate ruler for measuring objects.

Summary for the Non-Mathematician

The Subject: Counting specific types of numbers (Smooth and Rough).
The Challenge: Our counting formulas are slightly off (the Error Term).
The Discovery: The errors in Smooth numbers and Rough numbers are mathematically tethered together.
The Result: By linking them, we can use our knowledge of one to fix our mistakes in the other, leading to much more precise mathematical predictions.

This paper, titled "A Link Between Error Terms When Counting Smooth and Rough Numbers" by Andreas Weingartner, establishes a formal mathematical relationship between the error terms associated with two fundamental counting functions in analytic number theory: the count of $y$ -smooth numbers ( $\Psi(x, y)$ ) and the count of $y$ -rough numbers ( $\Phi(x, y)$ ).

1. The Problem

In number theory, we often seek to estimate:

$\Psi(x, y)$ (Smooth numbers): The number of integers up to $x$ whose prime factors are all $\le y$ . The standard approximation is $x\rho(u)$ , where $\rho(u)$ is the Dickman function.
$\Phi(x, y)$ (Rough numbers): The number of integers up to $x$ whose prime factors are all $> y$ . The standard approximation involves the Buchstab function $\omega(u)$ .

While precise approximations exist for both (notably de Bruijn’s $\Lambda(x, y)$ for smooth numbers and various forms of $W(x, y)$ for rough numbers), the error terms—the differences between the actual counts and these approximations—have historically been studied somewhat independently. The problem addressed here is to find a functional link that allows error bounds from one domain to be transferred to the other.

2. Methodology

The author employs Laplace transforms and Möbius inversion to bridge the two functions. The methodology follows these steps:

Definition of Error Terms: The author defines specific error terms:
- $\Delta(x, y)$ for smooth numbers.
- $Q(x, y)$ and $Q^*(x, y)$ for rough numbers.
- $R(x, y)$ and $R^*(x, y)$ , which are sums of the rough-number error terms over the set of smooth numbers.
Integral Equations: By expressing the counting functions through integral equations (specifically using the properties of $\rho(u)$ and $\omega(u)$ ), the author derives equations in the Laplace domain.
Inversion: By inverting these Laplace transforms, the author establishes identities that express the smooth-number error $\Delta$ in terms of the rough-number error $R$ , and vice versa.

3. Key Contributions and Results

The core of the paper is Theorem 1, which provides four explicit identities. These identities show that the error in counting smooth numbers is essentially a convolution of the error in counting rough numbers with the derivative of the Dickman function.

The main results include:

Theorem 1 (The Link): Provides the bidirectional formulas:
- $\Delta(x, y)$ can be expressed as an integral involving $R(x, y)$ .
- $R(x, y)$ can be expressed as an integral involving $\Delta(x, y)$ .
Corollary 1 & 3 (Transfer of Bounds): These establish that if one error term is bounded by a certain function $f(y)$ , the other error term is also bounded by a related function (scaled by the product $\Pi(y)^{-1}$ ).
Corollary 2 (Explicit Bounds): This is a major practical application. By using known explicit bounds for rough numbers (from Fan), the author derives new, explicit upper bounds for the smooth-number error term $\Delta(x, y)$ $Δ (x, y)$ .
- Unconditional bound: $\Delta(x, y) \ll x \frac{\log^{1/4} y}{\exp(\sqrt{\log y / 6.315})}$ .
- Conditional bound (under RH): $\Delta(x, y) \ll x \frac{\log^2 y}{\sqrt{y}}$ .

4. Significance

The significance of this work is both theoretical and practical:

Theoretical Unity: It demonstrates that the distribution of smooth numbers and rough numbers is not merely "similar" but is mathematically dual. The error in one is intrinsically tied to the error in the other through the structure of the integers (specifically via the property that every integer is a product of a smooth and a rough part).
Computational Utility: It provides a "shortcut" for researchers. Instead of performing the difficult task of bounding $\Delta(x, y)$ directly, one can use existing, highly optimized bounds for $\Phi(x, y)$ and apply the author's formulas to obtain precise estimates for smooth numbers.
Refinement of Estimates: The paper provides sharper, explicit constants for error terms, which are crucial for applications in cryptography and computational number theory where precise bounds on the density of smooth numbers are required.