Dickman Constants in Asymptotic Analysis

• Dickman constants are arithmetic coefficients that quantify corrections in the asymptotic expansion of the Dickman function, which measures the density of u-smooth numbers.
• They are defined via saddle-point and generating function methods, with explicit examples like C0 = e^γ/√(2π) and expressions involving ζ values.
• These constants play a key role in analytic number theory, probabilistic combinatorics, and even in analogies with quantum field theory through their impact on cycle length distributions and perpetuity models.

The Dickman constants are a family of arithmetic constants that govern the asymptotics of the Dickman function and its relatives, appearing pervasively in analytic number theory, random permutations, probabilistic combinatorics, and the theory of perpetuities. These constants emerge as coefficients in the asymptotic expansions of the proportion of integers free of large prime divisors, the density of cycle lengths in random permutations, and in the large-$u$ expansions of certain infinitely-divisible distributions tied to scale-invariant Poisson processes.

1. Classical Dickman Function and Its Expansions

The Dickman function $\rho(u)$, for $u\geq1$, is defined via the delay-differential equation

$$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$

encoding the density of $u$-smooth numbers (integers with no prime factor exceeding $N^{1/u}$ for $N\to\infty$) (1004.05191005.3494). In analytic number theory, the large-$u$ behavior of $\rho(u)$ admits a precise saddle-point expansion: $$\rho(u) \sim C_0\,u^{-u}e^u \bigg(1 + \frac{C_1}{u} + \frac{C_2}{u^2} + \cdots \bigg),$$ with

$\rho(u)$0

where $\rho(u)$1 is Euler’s constant; $\rho(u)$2 are further Dickman constants. The constants $\rho(u)$3 are extracted via the expansion of the saddle-point in the Laplace method, with higher-order corrections encoded through derivatives of the cumulant generating function $\rho(u)$4 (cf. (Arratia et al., 2016)): $\rho(u)$5

The coefficients $\rho(u)$6 encode intricate corrections to the leading $\rho(u)$7 asymptotics and are expressible in terms of $\rho(u)$8 and explicit polynomials (Arratia et al., 2016).

2. Polylogarithmic Representation and Generating Functions

The Dickman constants also surface as the coefficients in the Maclaurin expansion of the exponential generating function (1004.05191005.3494Franze, 2017): $\rho(u)$9 This remarkable generating function links the Dickman expansion to the expansion

$u\geq1$0

(up to normalization), where $u\geq1$1 is the Riemann zeta function. As a result, each $u\geq1$2 is a $u\geq1$3-linear combination of products of $u\geq1$4 with $u\geq1$5.

Explicit closed forms are provided for small $u\geq1$6: $u\geq1$7

$u\geq1$8

with recursive and combinatorial formulas via Bell polynomials (Franze, 2017).

Soundararajan and Broadhurst (1005.34941004.0519) established explicit integral formulas: $u\geq1$9 enabling systematic computation and providing a bridge to generalized polylogarithms.

3. Dickman Constants in Probabilistic and Perpetuity Expansions

From a probabilistic perspective, the Dickman constants appear in the large-$$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$0 expansion of the tail and density of the Dickman distribution (arising as a perpetuity): $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$1 where $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$2 solves $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$3 (Iksanov et al., 15 Apr 2026). The pre-factor $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$4 is often named the "Dickman constant" in analytic number theory.

This connection is formalized by representing the Dickman variable as the sum of arrivals in a scale-invariant Poisson process (Arratia et al., 2016), and by leveraging Cramér tilting and saddle-point techniques. The correction terms to the leading $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$5 law are again dictated by the sequence of Dickman constants.

4. Asymptotic, Arithmetic, and Combinatorial Structure

The Dickman constants enjoy explicit arithmetic structure. Each $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$6 can be written as a rational linear combination of monomials in $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$7 (Riemann zeta values) via combinatorial (Bell polynomial) representations (Franze, 2017). The leading-order term $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$8 is transcendental; subsequent coefficients involve only standard special values and their products.

Numerically, the first few Dickman constants are: | $$\rho(u)=1 \text{ for } 0\le u\le 1, \qquad u\rho'(u) + \rho(u-1) = 0 \text{ for } u>1,$$9 | $u$0 | Approximate Value | |-----|----------------------|--------------------------| | 0 | $u$1 | $u$2 | | 1 | $u$3 | $u$4 | | 2 | $u$5 | $u$6 | | 3 | $u$7 | $u$8 | | 4 | $u$9 | $N^{1/u}$0 | | 5 | $N^{1/u}$1 | $N^{1/u}$2 |

All higher $N^{1/u}$3 vanish modulo products of zeta values; there are no new transcendental constants at any order (1004.05191005.3494Franze, 2017).

5. Generalizations: Golomb–Dickman Constant and Ewens Measure

The generalized Golomb–Dickman constant $N^{1/u}$4 arises as the limiting expected relative size of the longest cycle in Ewens-distributed random permutations on $N^{1/u}$5 with parameter $N^{1/u}$6: $N^{1/u}$7 where $N^{1/u}$8 is the length of the longest cycle. An explicit integral representation is given by (Mendonça et al., 24 Mar 2026): $N^{1/u}$9 with $N\to\infty$0 the exponential integral. The case $N\to\infty$1 recovers the classical Golomb–Dickman constant $N\to\infty$2.

As $N\to\infty$3, $N\to\infty$4, reflecting the predominance of a giant cycle; as $N\to\infty$5, $N\to\infty$6, indicating fragmentation into many small cycles. This framework unifies the analysis for all regime transitions between long-cycle and many-small-cycle dominance in permutation groups.

6. Connections, Open Problems, and Physics Analogies

The Dickman constants exhibit deep analogies with structures in perturbative quantum field theory (QFT) and condensed matter: constants governing higher-order corrections in QFT (e.g., splitting functions in QCD) are also expressible in terms of multiple zeta values (MZVs) and polylogarithms of a similar type. Notably, possible "irreducible" MZV contributions at certain weights are absent from Dickman constants, mirroring patterns in spin-chain computations (Broadhurst, 2010).

Current open problems include proving the full generating function conjecture, deeper analytic understanding of the absence of certain MZVs, and exploration of whether all relevant constants can be reduced to products of simple zeta values (1005.34941004.0519).

7. Summary Table: Key Definitions and Formulae

Context	Definition / Formula	Reference
Dickman constant $N\to\infty$7	$N\to\infty$8	(Arratia et al., 2016)
Dickman constants $N\to\infty$9	Maclaurin coefficients of $u$0	(Broadhurst, 2010)
Generating function	$u$1	(Soundararajan, 2010)
Golomb–Dickman $u$2	$u$3	(Mendonça et al., 24 Mar 2026)
Leading asymptotic	$u$4	(Arratia et al., 2016)
Cycle distributions	$u$5	(Mendonça et al., 24 Mar 2026)

The Dickman constants function as a central organizing object across several domains, encoding the subtle arithmetic and probabilistic corrections in the asymptotics of fundamental counting problems, probabilistic combinatorics, and perpetuities (1005.34941004.0519Franze, 2017Mendonça et al., 24 Mar 2026).