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Explicit bounds for Buchstab's function

This paper establishes numerically explicit and easily evaluable upper and lower bounds for Buchstab's function $\omega(u)$ , eliminating the need for numerical solutions of its defining delay differential equation.

Original authors: Andreas Weingartner

Published 2026-07-27

📖 4 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

Imagine you are a detective trying to count the number of "lonely" integers in a massive crowd. In the world of numbers, some integers are social butterflies, made up of many small prime factors (like 12, which is 2 × 2 × 3). Others are the introverts, the "loners" that refuse to have any prime factors smaller than a certain size. Mathematicians call these numbers "y-smooth" or "y-rough" depending on the context, but the core mystery is: how many of these loners exist up to a certain point?

To solve this, mathematicians use a special tool called Buchstab's function, denoted as $\omega(u)$ . Think of this function as a weather vane that spins wildly at first but eventually settles into a steady, predictable breeze. This breeze represents a specific mathematical constant (related to Euler's constant, $\gamma$ ). The function tells us exactly how the distribution of these "loner" numbers behaves as we look at larger and larger ranges. Why do we care? Because the behavior of these numbers is deeply tied to the distribution of prime numbers themselves. If we can predict how these loners behave, we get a sharper lens for seeing where prime numbers hide, which is crucial for everything from pure math theory to the security of digital codes.

For a long time, calculating this function was like trying to predict the weather by solving a complex, time-delayed puzzle. You had to solve a specific "delay differential equation," which is a fancy way of saying the value of the function right now depends on what it was a moment ago, creating a chain reaction that is very hard to compute by hand or even by computer without a lot of heavy lifting.

This paper, written by Andreas Weingartner, steps in with a fresh, much simpler approach. Instead of forcing the reader to solve that difficult, time-delayed puzzle every time they want a value, the author provides a set of "explicit bounds." Imagine you are trying to guess the height of a growing plant. Instead of measuring the plant every single second and calculating its growth rate from scratch, Weingartner gives you a highly accurate ruler and a simple formula. He shows that for any number $u$ (where $u$ represents the scale of the problem), you can calculate the value of the function using a specific formula involving complex numbers and a special function called the Lambert W function, and then simply check if your answer falls within a very tight "error margin."

The paper proves that this new method works with incredible precision. For values of $u$ between 3 and 6, the error in the estimate is guaranteed to be smaller than $\frac{1}{2u(u-1)}$ . For larger numbers ( $u \ge 6$ ), the error shrinks even further, staying below $\frac{1}{12u \log u}$ . The author doesn't just guess these limits; he rigorously proves them using a technique called the "saddle point method," which is like finding the highest peak on a mountain range to get the best view of the surrounding landscape. He also provides a second, even more precise version of the formula for larger numbers, reducing the error to a tiny fraction involving $u^{-2}$ or even $u^{-3}$ .

The beauty of this work is that it turns a difficult, iterative calculation into a straightforward, one-step recipe. You plug in your number, do the math (which involves standard functions found in most computer software), and you get a result that is guaranteed to be within a specific, tiny distance from the true answer. The paper includes a table showing these calculations for numbers like 6, 10, 40, and even 10,000, demonstrating that the new formulas match the "true" values perfectly. It's a bit like replacing a complex, winding maze with a straight highway: you get to the same destination, but you get there faster, with a clear map, and you know exactly how close you are to the finish line.

Technical Summary: Explicit Bounds for Buchstab's Function

Problem Statement
Buchstab's function, $\omega(u)$ , characterizes the distribution of integers free of small prime factors. Specifically, for a fixed $u > 1$ , the count of integers up to $x$ with all prime factors greater than $x^{1/u}$ , denoted $\Phi(x, x^{1/u})$ , is asymptotically $x\omega(u)/\log(x^{1/u})$ . As $u \to \infty$ , $\omega(u)$ converges rapidly to $e^{-\gamma}$ (where $\gamma$ is Euler's constant), oscillating around this limit.

The behavior of these oscillations is critical in analytic number theory, particularly in establishing bounds for the number of primes in short intervals. Maier [5] demonstrated that the lim sup and lim inf of the density of primes in intervals of length $(\log x)^v$ depend directly on the maximum and minimum values of $W(u) := \omega(u) - e^{-\gamma}$ for $u \ge v$ . While Hildebrand [3] provided an asymptotic formula for $W(u)$ involving complex saddle-point analysis, the error term in that approximation was not numerically explicit. The objective of this paper is to establish rigorous, numerically explicit upper and lower bounds for $W(u)$ that are easy to evaluate without solving the underlying delay differential equation $(u\omega(u))' = \omega(u-1)$ numerically.

Methodology
The author employs the saddle-point method applied to the inverse Laplace transform of Buchstab's function. The Laplace transform is given by $\hat{\omega}(s) = e^{J(s)} - 1$ , where $J(s)$ is the exponential integral function.

Saddle Point Definition: A unique complex solution $\zeta = \mu - i\eta$ is defined for the equation $e^\zeta = -u\zeta$ within the range $\mu \ge 1$ and $0 < \eta < 3\pi/2$ . This $\zeta$ serves as the saddle point for the integral representation of $W(u)$ .
Taylor Expansion: Unlike previous works that used lower-order expansions, this paper utilizes a Taylor polynomial of order seven near the saddle point to approximate the integrand. This higher-order expansion allows for tighter control over the error terms.
Complex Analysis: The analysis involves estimating the real and imaginary parts of $J(-\zeta + i\tau)$ and its derivatives. The author derives explicit bounds for these derivatives (Lemma 4) and establishes decay estimates for the integrand away from the saddle point (Lemma 5).
Error Decomposition: The integral for $W(u)$ is split into a main term (centered at the saddle point) and tail contributions. The tail contributions are bounded using explicit inequalities derived from integration by parts and contour integration estimates.
Numerical Verification: For small values of $u$ (specifically $u \le 103$ ), the theoretical bounds are verified by numerically calculating $W(u)$ using the algorithm of Marsaglia, Zaman, and Marsaglia [6] and comparing the results against the derived error bounds.

Key Contributions and Results
The paper presents three theorems providing increasingly precise approximations for $W(u)$ with explicit error bounds.

Theorem 1: Provides a primary approximation using the function $\Phi_1(u) = \Phi(u)(1 - \frac{1}{12u})$ .
$W(u) = 2|\Phi_1(u)| \{ \cos(\arg(\Phi_1(u))) + \theta_1(u) \}$
The error term satisfies $|\theta_1(u)| < \frac{1}{2u(u-1)}$ for $3 \le u \le 6$ and $|\theta_1(u)| < \frac{1}{12u \log u}$ for $u \ge 6$ .
Theorem 2: Offers a more precise approximation using $\Phi_2(u) = \Phi(u)(1 + \alpha(u))$ , where $\alpha(u)$ is a specific rational function of $\zeta$ and $u$ derived from the fourth and fifth derivatives of $J$ .
$W(u) = 2|\Phi_2(u)| \{ \cos(\arg(\Phi_2(u))) + \theta_2(u) \}$
The error term satisfies $|\theta_2(u)| < 420u^{-6}$ for $6 \le u \le 19$ and $|\theta_2(u)| < 0.005u^{-2}$ for $u \ge 16$ .
Theorem 3: Presents an even smaller error term by including a second correction term $\beta(u)$ in the main factor, defining $\Phi_3(u) = \Phi(u)(1 + \alpha(u) + \beta(u))$ .
$W(u) = 2|\Phi_3(u)| \{ \cos(\arg(\Phi_3(u))) + \theta_3(u) \}$
The error term satisfies $|\theta_3(u)| < 0.01u^{-3}$ for $18 \le u \le 103$ and $|\theta_3(u)| < 8.4u^{-3}$ for $u \ge 103$ .

The paper notes that $\Phi(u)$ can be computed using standard special functions: the incomplete Gamma function for $J(s)$ and the Lambert $W$ function (specifically the $k=1$ branch) for $\zeta$ . Table 1 in the paper illustrates the high precision of these bounds by comparing them to numerically computed values for various $u$ .

Significance and Claims
The paper claims to provide the first numerically explicit upper and lower bounds for Buchstab's function that do not require the numerical solution of the delay differential equation. The methodology mirrors previous work by the author on Dickman's function [9], adapting real-valued estimates to the complex domain required by Buchstab's function.

The author modestly frames the significance of these results as establishing "numerically explicit" estimates. The paper does not claim to solve the problem of prime distribution in short intervals but rather provides the necessary explicit bounds on $W(u)$ that allow such problems to be addressed with concrete numerical constants. The error terms are shown to be significantly smaller than the $O(1/u)$ error term found in Hildebrand's earlier asymptotic formula [3], particularly for moderate values of $u$ . The results are validated through a combination of rigorous analytic bounds for large $u$ and direct numerical computation for smaller $u$ .

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