Dickman–de Bruijn Function Overview

• Dickman–de Bruijn function is defined as the unique continuous solution to a differential-difference equation, representing the limiting density of friable (smooth) numbers.
• Its asymptotic expansion, derived using saddle-point techniques and polylogarithmic integrals, shows rapid decay, with p(u) diminishing faster than any exponential function.
• Generalizations of the function extend its framework to multiple integrals and sieve theory, offering insights into prime factor distributions and applications in cryptography.

The Dickman–de Bruijn function, denoted $\rho(u)$ or $p(u)$, occupies a central position in analytic number theory as the function describing the limiting density of friable (or smooth) numbers—that is, positive integers up to $x$ whose prime factors are all at most $y$ for $y = x^{1/u}$. Beyond its foundational role in understanding the distribution of smooth numbers, this function and its generalizations are deeply intertwined with the structure of delay-differential equations, sieve theory, asymptotic analysis, and connections to special functions such as the gamma and zeta functions.

1. Definition and Core Properties

The Dickman–de Bruijn function $p(u)$ is defined for $u \geq 0$ as the unique continuous solution to the differential-difference equation

$$p'(u) = -\frac{p(u-1)}{u}, \quad u > 1;\qquad p(u) = 1, \quad 0 \leq u \leq 1.$$

Equivalently, it admits an integral formulation: $$p(u) = \begin{cases} 1, & 0 \leq u \leq 1, \ \frac{1}{u} \int_{u-1}^u p(t)\,dt, & u > 1. \end{cases}$$ The function is continuous, strictly decreasing for $u > 1$, and satisfies $0 < p(u) < 1$ for all $u > 0$. For $1 < u < 2$, $p(u) = 1 - \log u$. Its rapid decay at large $u$ is such that $p(u) < 1/\Gamma(u+1)$, decaying faster than any negative exponential or negative power of $u$.

A crucial probabilistic interpretation is that $p(u)$ describes the limiting density of positive integers $n \leq x$ with all prime factors $\leq x^{1/u}$: $$\lim_{x \to \infty} \frac{\Psi(x, x^{1/u})}{x} = p(u), \quad \Psi(x, y) = |\{n \leq x \mid P(n) \leq y \}|$$ where $P(n)$ is the largest prime factor of $n$.

2. Asymptotic Expansions and Analytic Structure

2.1. Classical Asymptotics

De Bruijn furnished the pivotal asymptotic expansion for $p(u)$ as $u \to \infty$: $$p(u) = \exp \left\{ -u \left( \log u + \log_2 u - 1 + \frac{\log_2 u - 1}{\log u} + O\left( \left( \frac{\log_2 u}{\log u} \right)^2 \right) \right) \right\}$$ where $\log_2 u = \log\log u$ (Moree, 2012). This demonstrates that $p(u)$ decays much faster than any exponential in $u$.

2.2. Polylogarithmic Expansions

Further analytic structure appears in asymptotic expansions involving polylogarithm-type functions. These include Dickman polylogarithms $L_k(t)$ and iterated logarithmic integrals $I_k(u)$ defined recursively, establishing expansions for $p(u)$ and its relatives: $$p(u) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} I_k(u),$$ with

$I_0(u) = 1, \quad I_1(u) = \log u,$

and for $k \geq 1$,

$I_k(u) = \int_1^u \frac{I_{k-1}(t)}{t} dt.$

Asymptotically, as $u \to \infty$: $$I_k(u) = \sum_{j=0}^k D_j (\log u)^{k-j} + O\left( \frac{(\log u)^k}{u} \right),$$ where $D_j = (-1)^j j! C_j$ and $(C_j)$ are the Dickman constants described below (Soundararajan, 2010). The corresponding expansion for polylogarithms $L_k(t)$ as $t \to 0$ is

$$L_k(t) = \sum_{j=0}^{k} \frac{C_j}{(k - j)!} (\log t)^{k - j} + O(t |\log t|).$$

3. Dickman Constants, Generating Functions, and Arithmetic Structure

A sequence of "Dickman constants" $(C_k)$ appears as coefficients in the above expansions and possesses a rich arithmetic structure. The generating function for these constants is

$$\sum_{k=0}^\infty C_k z^k = \frac{e^{\gamma z}}{\Gamma(1-z)},$$

where $\gamma$ is Euler's constant and $\Gamma$ is the gamma function (Soundararajan, 2010, Franze, 2017). This generating function encodes the arithmetic and combinatorial complexity of $C_k$, which, for each $k \geq 1$, can be expressed in terms of Bell polynomials and values of the Riemann zeta function: $$C_r = \frac{1}{r!} \sum_{k=1}^r (-1)^k B_{r,k}(0, 1! \zeta(2), 2! \zeta(3), \ldots).$$ Generalizations appear in higher-order Dickman–de Bruijn functions (see Section 4).

Explicit contour integral representations are given: $$C_k = \frac{1}{k! 2\pi i} \int_{c - i\infty}^{c + i\infty} e^{s} (\log s + \gamma)^k \frac{ds}{s^2}.$$ The significance of this analytic structure is twofold: it not only organizes the expansion coefficients arithmetically, but also links them to special functions and allows for precise analytic continuation.

4. Generalizations: Dickman–de Bruijn Families and Multiple Integrals

The Dickman–de Bruijn function admits broad generalizations, yielding functions $\rho_\kappa(u)$ defined for positive integers $\kappa$ by: $$\left(u^{1-\kappa} \rho_{\kappa}(u)\right)' = -\kappa u^{-\kappa} \rho_{\kappa}(u-1), \quad u > 1;\qquad \rho_{\kappa}(u) = \frac{u^{\kappa-1}}{(\kappa-1)!} \quad (0 < u \leq 1),$$ with $\rho_\kappa(u) = 0$ for $u \leq 0$ (Franze, 2017).

These are represented through families of multiple integrals $K_\ell(u,\kappa)$: $$K_\ell(u, \kappa) = \frac{1}{\ell!} \int_{\substack{t_1,\ldots,t_\ell \ge 1 \ t_1+\cdots+t_\ell\le u}} (u - (t_1 + \cdots + t_\ell))^\kappa \frac{dt_1}{t_1} \cdots \frac{dt_\ell}{t_\ell},$$ with a decomposition

$$\rho_{\kappa}(u) = \sum_{0 \leq \ell < u} \frac{(-\kappa)^\ell}{(\kappa-1)!} K_\ell(u, \kappa-1),$$

which allows the corresponding expansions of $K_\ell(u, \kappa)$ to yield asymptotic expansions for $\rho_\kappa(u)$. For $\kappa=0$, this recovers the classical Dickman function and its properties; the general case is relevant in higher-dimensional sieve theory and analysis of more intricate friable integer sets.

The main asymptotic expansion (for fixed $\ell$ and $u \to \infty$) is: $$K_{\ell}(u, \kappa) = \sum_{m=0}^{\min(\kappa,\ell)} \sum_{n=m}^\kappa \sum_{r=0}^{\ell-m} D_{m,n,r,\kappa,\ell}\ u^{\kappa-n} \log^{\ell-m-r} u + O_{\kappa,\ell}\left(\frac{\log^\ell u}{u}\right),$$ where the $D_{m,n,r,\kappa,\ell}$ have explicit arithmetic expressions involving gamma, Dickman constants, and combinatorial numbers. The generalized Dickman constants $C_{r,\kappa}$ enjoy a comparable generating function: $$\sum_{r=0}^{\infty} C_{r,\kappa} z^r = \frac{e^{\gamma z}}{\Gamma(\kappa+1-z)}.$$ This arithmetic structure extends the deep connection to special values of the gamma and zeta functions established in the classical case.

5. Foundational Contributions and Analytic Techniques

De Bruijn's analytic methods, particularly his adaptation of the saddle-point technique, yielded the first precise error-controlled asymptotic for $p(u)$. The approach includes the introduction of a function $\xi(u)$, the unique positive solution of $(e^\xi - 1)/\xi = u$, which acts as the saddle-point parameter, and leads to precise global expansions: $$p(u) \sim \exp \left(-\xi(u) u + \int_1^u \xi(t)\,dt\right).$$ De Bruijn also computed the Laplace transform: $$\int_0^\infty p(u) e^{-s u} du = \exp\left(\gamma - \int_0^s \frac{1-e^{-t}}{t}dt\right),$$ where $\gamma$ is Euler's constant (Moree, 2012).

The paper of $p(u)$, its generalizations, and related integrals is structured by connections to linear functionals and differential-delay equations, often of the form

$u f'(u) + a f(u) + b f(u-1) = 0,$

where specific values of $a$ and $b$ recover the Dickman class of functions and various auxiliary functions for sieve methods.

Recent advances establish a rigorous analytic link between these expansions, numerical conjectures from the physics literature, and the arithmetic structure of their coefficients (Soundararajan, 2010, Franze, 2017).

6. Applications and Broader Context

The Dickman–de Bruijn function controls the statistical behaviour of friable numbers, and therefore underpins effective estimates in sieve theory, cryptography (integer factoring run time probabilistics), and the probabilistic structure of random factorizations. It appears in the paper of cycle lengths in random permutations, in forms such as the probability that the largest cycle is at most $n/u$ in an $n$-element random permutation (Golomb–Dickman constant).

Applications extend to precise estimates of

• the proportion of smooth numbers in given intervals,
• the distribution of the largest prime factor of a random integer,
• the analytic structure underlying sieve auxiliary functions,
• error terms in the Turán-Kubilius and related inequalities,
• and analysis of arithmetic functions in random and combinatorial models.

The explicit expansions and arithmetic formulae for the Dickman constants, and their generalizations, facilitate practical computations relevant for theoretical and computational analytic number theory.

7. Summary of Core Formulas

Context	Formula	Parameters/Description
Integral Definition	$p(u) = 1$ for $0\leq u\leq 1$; $p(u) = \frac{1}{u} \int_{u-1}^u p(t) dt$ for $u>1$	$u \geq 0$
Differential-Difference Equation	$p'(u) = -\frac{p(u-1)}{u}$, $u > 1$	$p(u) = 1$ for $u \leq 1$
Classical Asymptotic	$$p(u) = \exp\left\{-u[\log u + \log_2 u - 1 + o(1)]\right\}$$	$u \to \infty$
Laplace Transform	$$\int_0^\infty p(u) e^{-s u} du = \exp(\gamma - \int_0^s \frac{1-e^{-t}}{t}dt)$$	$\gamma$ is Euler’s constant
Dickman Constants Generating Fn.	$$\sum_{k=0}^\infty C_k z^k = \frac{e^{\gamma z}}{\Gamma(1-z)}$$	$C_k$: Dickman constants
Generalized Constants	$$\sum_{r=0}^\infty C_{r,\kappa} z^r = \frac{e^{\gamma z}}{\Gamma(\kappa+1-z)}$$	See above, $\kappa\in\mathbb{N}$

These formulae encapsulate the essential structure, expansion, and analytic phenomena associated with the Dickman–de Bruijn function and its generalizations. The underlying arithmetic, analytic, and combinatorial properties are fundamental to large segments of analytic number theory and its applications (Moree, 2012, Soundararajan, 2010, Franze, 2017).