Golomb-Dickman constant $\lambda$

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Number

$\lambda$

0.6243299885435508709929363831008372441796426201805292869735519024956380888551132544624602761955398689

Definition

Let $M_n$ be the length of the longest cycle in a uniformly random permutation of $n$ elements. The Golomb-Dickman constant $\lambda$ is the limiting mean proportion $\lambda=\lim_{n\to\infty} \mathbb E(M_n)/n$ [1] [3].

Formulas

(1)

$\lambda=\int_0^1 e^{\operatorname{li}(x)}\,dx=\int_0^\infty e^{-x-E_1(x)}\,dx=\int_0^\infty e^{-x+\operatorname{Ei}(-x)}\,dx$, where $\operatorname{li}$ is the logarithmic integral and $E_1(x)=-\operatorname{Ei}(-x)$ for $x>0$ [3] [4] [6].

(2)

$\lambda=\int_0^\infty \frac{\rho(u)}{(u+1)^2}\,du=1-\int_1^\infty \frac{\rho(u)}{u^2}\,du$, where $\rho$ is the Dickman-de Bruijn function [2].

Comments

(3)

The same constant occurs in number theory: the average number of digits in the largest prime factor of a random integer with $d$ digits is asymptotic to $\lambda d$ [4].

(4)

No rational or algebraic closed form is known for $\lambda$ [5].

Programs

(P1)

Python

import mpmath

mpmath.mp.dps = 120
f = lambda x: mpmath.exp(-x - mpmath.e1(x))
mpmath.quad(f, [0, 1, mpmath.inf])

References

[1]

L. A. Shepp and S. P. Lloyd, Ordered cycle lengths in a random permutation, Transactions of the American Mathematical Society 121 (1966), 340-357. (doi)

[2]

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bulletin of the American Mathematical Society 50 (2013), 527-628. (arXiv) (doi)

Links

[3]

MathWorld: Golomb-Dickman Constant

[4]

OEIS A084945: decimal expansion of the Golomb-Dickman constant

[5]

Wikipedia: Golomb-Dickman constant

[6]

DLMF 6.6: Power series for exponential integrals

Similar tables

Values of the logarithmic integral — appears in the integral representation (1)

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: yes (it holds the single named constant $\lambda$)

How they were obtained:

The attached generator computes (2) with an interval Taylor method for the Dickman-de Bruijn function $\rho$. On each unit interval it carries $\rho$ as a midpoint Taylor polynomial with ball coefficients, applies $u\rho'(u)=-\rho(u-1)$ coefficient by coefficient, and includes explicit geometric tails from the series for $1/u$ and $(u+1)^{-2}$. The tail after the last computed interval is bounded by monotonicity of $\rho$.

The stored ball was also compared with the decimal expansion in OEIS A084945 [4] and with direct mpmath quadrature of (1) at 150 decimal digits.