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Explicit bounds for Dickman's function

This paper establishes numerically explicit upper and lower bounds for Dickman's function $\rho(u)$ that achieve a relative error of less than $0.005/u^2$ for all $u \ge 5$ , enabling accurate approximate evaluation without solving the underlying delay differential equation.

Original authors: Andreas Weingartner

Published 2026-06-09

📖 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 trying to count a very specific type of number in the vast ocean of all integers. Specifically, you want to know how many numbers exist where their biggest "building block" (prime factor) isn't too huge compared to the number itself.

Mathematicians have a special tool for this, a function named Dickman's function (let's call it $\rho$ ). Think of $\rho$ as a weather forecast for these numbers. It tells you the "density" or probability of finding these special numbers.

However, calculating this forecast is notoriously difficult. The standard way to get the answer is like trying to predict the weather by solving a complex, moving puzzle called a "delay differential equation." You have to solve it step-by-step, and if you make a tiny mistake early on, the whole forecast gets messy. It's slow, tedious, and requires a supercomputer to get it right.

What this paper does:
Andreas Weingartner wrote this paper to give us a much simpler, highly accurate shortcut to predict this weather without solving the hard puzzle.

Here is the breakdown of his discovery using simple analogies:

1. The "Cheat Sheet" (The Approximation)

Mathematicians already knew a "rough draft" formula (called $\tilde{\rho}$ ) that gets close to the real answer as the numbers get bigger. But a rough draft isn't good enough if you need precision.

Weingartner created a high-precision "Cheat Sheet."

The Old Way: You had to run a marathon (solve the complex equation) to get the answer.
The New Way: You just look up a value on a list and plug it into a simple calculator formula.

2. The "Error Margin" (How good is the shortcut?)

The paper proves that this shortcut is incredibly reliable.

Imagine you are measuring the height of a mountain. The old rough draft might be off by a few feet.
Weingartner's new formula is so precise that the error is less than 0.005% for numbers larger than 5.
To use a metaphor: If you were measuring the distance from the Earth to the Moon, this formula would be off by less than the width of a human hair.

3. The "Safety Net" (Upper and Lower Bounds)

The paper doesn't just give one number; it gives a range (a "sandwich").

It says: "The real answer is definitely higher than this lower number, and lower than this upper number."
The gap between these two numbers is so tiny that for all practical purposes, you can just pick the middle value and be correct.

4. The "Magic Ingredients"

To build this shortcut, the author used some advanced mathematical tools (like the Lambert W function and the Exponential Integral), but you don't need to know what they are. Think of them as specialized lenses.

The author looked at the problem through these lenses and found a pattern.
He proved that if you use a specific correction factor (which he calls $\alpha(u)$ ), the shortcut becomes almost perfect.

5. The "Proof" (Why should we trust it?)

The author didn't just guess; he did the heavy lifting to prove it works.

He broke the problem down into tiny pieces (like slicing a loaf of bread).
He checked the edges of the loaf to make sure nothing was falling off.
He used a computer to verify the results for smaller numbers (where the shortcut is hardest to use) and found that his formula matched the "gold standard" calculations perfectly.

Summary

In everyday terms, this paper is like giving a driver a GPS navigation system instead of a paper map with complex, hand-drawn routes.

Before: You had to navigate a winding, confusing road (solving the differential equation) to get to your destination (the value of $\rho(u)$ ).
Now: You just type in your destination, and the GPS gives you the exact distance and time with a guarantee that you won't be off by more than a few inches.

The paper provides a way to calculate this specific mathematical function quickly, easily, and with a guarantee of extreme accuracy, saving mathematicians from having to do the heavy, slow lifting every time they need the answer.

Technical Summary: Explicit Bounds for Dickman's Function

Problem Statement
Dickman's function, $\rho(u)$ , is defined as the continuous solution to the delay differential equation $u\rho'(u) + \rho(u-1) = 0$ with the initial condition $\rho(u) = 1$ for $0 \le u \le 1$ . It represents the natural density of integers $n$ whose largest prime factor does not exceed $n^{1/u}$ . While the asymptotic behavior of $\rho(u)$ as $u \to \infty$ is well-understood via the work of de Bruijn and Alladi, these results typically involve implicit error terms (e.g., $O(1/u)$ ). The primary challenge addressed in this paper is to establish numerically explicit upper and lower bounds for $\rho(u)$ that allow for approximate evaluation without solving the delay differential equation numerically.

Methodology
The author employs the saddle point method applied to the inverse Laplace integral representation of $\rho(u)$ . The approach refines the proof technique found in Tenenbaum's work, which established Alladi's result with a relative error of $O(u^{-1})$ . To achieve the tighter explicit bounds required here, the methodology involves:

Precise Error Tracking: Keeping track of all constant factors rather than absorbing them into big-O notation.
Refined Integration: Dividing the range of integration into additional intervals to isolate contributions more accurately.
Higher-Order Expansion: Utilizing a Taylor polynomial of order seven (instead of three) near the saddle point to capture higher-order terms in the expansion of the integrand.
Explicit Lemmas: Deriving specific inequalities for the derivatives of the integral function $I(s) = \int_0^s \frac{e^z-1}{z} dz$ and bounding the tail contributions of the integral using lemmas related to the incomplete gamma function.

The function $\tilde{\rho}(u)$ , serving as the primary approximation, is defined as:
$\tilde{\rho}(u) := \frac{\exp\{\gamma - u\xi + I(\xi)\}}{\sqrt{2\pi I''(\xi)}}$
where $\xi = \xi(u)$ is the unique positive solution to $e^\xi = 1 + u\xi$ , and $\gamma$ is Euler's constant.

Key Contributions and Results
The paper establishes two main theorems providing explicit bounds and estimates for $\rho(u)$ :

Theorem 1 (Explicit Bounds): For $u > 1$ , the function $\rho(u)$ is bounded by:
$\tilde{\rho}(u) \left(1 - \frac{1}{12u}\right) < \rho(u) < \tilde{\rho}(u) \left(1 - \frac{1}{12u(1 + \frac{1}{\log u})}\right)$
This provides a convenient range for $\rho(u)$ using only elementary functions and the Lambert $W$ function (to solve for $\xi$ ).
Theorem 2 (Refined Estimate with Error Term): For $u > 1$ , the function can be expressed as:
$\rho(u) = \tilde{\rho}(u) \left(1 + \alpha(u) + \frac{\theta(u)}{u^2}\right)$
where $\alpha(u)$ is an explicit expression involving the derivatives of $I(\xi)$ :
$\alpha(u) := \frac{1}{8}\frac{I^{(4)}(\xi)}{(I''(\xi))^2} - \frac{5}{24}\frac{(I^{(3)}(\xi))^2}{(I''(\xi))^3}$
Crucially, the error term is bounded explicitly: $|\theta(u)| < 0.005$ for all $u \ge 5$ . This results in a relative error of less than $0.005/u^2$ .

The paper also provides a method for the numerical calculation of $\rho(u)$ for verification:

For $1 \le u \le 50$ , it uses the Taylor polynomial algorithm of Marsaglia, Zaman, and Marsaglia.
For larger $u$ , it evaluates the inverse Laplace integral numerically, noting that the integrand converges to the standard normal density function.

Significance and Claims
The paper claims that these results allow for the "approximate evaluation of $\rho(u)$ without the need to solve the delay differential equation numerically." By providing explicit constants and error bounds, the work makes the asymptotic estimates of de Bruijn and Alladi "entirely explicit."

The author notes that while other works (Smida, Xuan) provide estimates with smaller error magnitudes, they lack numerical explicitness. The explicit bounds derived here are particularly useful for applications involving $\Psi(x, y)$ , the count of integers up to $x$ with largest prime factor $\le y$ . Specifically, combined with conditional results from Gorodetsky (assuming the Riemann Hypothesis), these bounds yield conditional lower bounds for $\Psi(x, y)$ that are "easy to evaluate numerically."

The paper validates its results by comparing them against values obtained by van de Lune and Wattel, who used the trapezoidal rule to approximate $\rho(u)$ to at least five significant digits for $u \le 1000$ . The values derived from Theorems 1 and 2 are consistent with these established numerical benchmarks.