Golomb–Dickman constant

In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is

${\displaystyle \lambda =0.62432998854355087099293638310083724\dots .}$

Definitions

Let a_n be the average — taken over all permutations of a set of size n — of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is

${\displaystyle \lambda =\lim _{n\to \infty }{\frac {a_{n}}{n}}.}$

In the language of probability theory, $\lambda n$ is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size n.

In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer. More precisely,

${\displaystyle \lambda =\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=2}^{n}{\frac {\log(P_{1}(k))}{\log(k)}},}$

where ${\displaystyle P_{1}(k)}$ is the largest prime factor of k. So if k is a d digit integer, then ${\displaystyle \lambda d}$ is the asymptotic average number of digits of the largest prime factor of k.

The Golomb–Dickman constant appears in number theory in a different way. What is the probability that second largest prime factor of n is smaller than the square root of the largest prime factor of n? Asymptotically, this probability is $\lambda$. More precisely,

${\displaystyle \lambda =\lim _{n\to \infty }{\text{Prob}}\left\{P_{2}(n)\leq {\sqrt {P_{1}(n)}}\right\}}$

where ${\displaystyle P_{2}(n)}$ is the second largest prime factor n.

Formulae

There are several expressions for $\lambda$. Namely,

${\displaystyle \lambda =\int _{0}^{\infty }e^{-t-E_{1}(t)}dt}$

where $E_1(t)$ is the exponential integral,

${\displaystyle \lambda =\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}dt}$

and

${\displaystyle \lambda =\int _{0}^{\infty }{\frac {\rho (t)}{(t+1)^{2}}}dt}$

where ${\displaystyle \rho (t)}$ is the Dickman function.

See also

• Random permutation
• Random permutation statistics

External links

• Weisstein, Eric W. "Golomb-Dickman Constant". MathWorld.

• "Sloane's A084945". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 284–286. ISBN 0-521-81805-2.