Golomb–Dickman constant

In mathematics, the Golomb–Dickman constant, named after Solomon W. Golomb and Karl Dickman, is a mathematical constant, which arises in the theory of random permutations and in number theory. Its value is

${\displaystyle \lambda =0.62432998854355087099293638310083724\dots }$ (sequence A084945 in the OEIS)

It is not known whether this constant is rational or irrational.^[1]

It's simple continued fraction is given by ${\displaystyle [0;1,1,1,1,1,22,1,2,3,1,...]}$, which appears to have an unusually large number of 1s.^[2]

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, ${\displaystyle \lambda n}$ 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 (sequence A006530 in the OEIS) . 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 ${\displaystyle \lambda }$. 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.

The Golomb-Dickman constant also arises when we consider the average length of the largest cycle of any function from a finite set to itself. If X is a finite set, if we repeatedly apply a function f: X → X to any element x of this set, it eventually enters a cycle, meaning that for some k we have ${\displaystyle f^{n+k}(x)=f^{n}(x)}$ for sufficiently large n; the smallest k with this property is the length of the cycle. Let b_n be the average, taken over all functions from a set of size n to itself, of the length of the largest cycle. Then Purdom and Williams^[3] proved that

${\displaystyle \lim _{n\to \infty }{\frac {b_{n}}{\sqrt {n}}}={\sqrt {\frac {\pi }{2}}}\lambda .}$

Formulae

There are several expressions for ${\displaystyle \lambda }$. These include:^[4]

${\displaystyle \lambda =\int _{0}^{1}e^{\mathrm {li} (t)}\,dt}$

where ${\displaystyle \mathrm {li} (t)}$ is the logarithmic integral,

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

where ${\displaystyle E_{1}(t)}$ is the exponential integral, and

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

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.
• OEIS sequence A084945 (Decimal expansion of Golomb-Dickman constant)
• Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 284–286. ISBN 0-521-81805-2.

References

1. Lagarias, Jeffrey (2013). "Euler's constant: Euler's work and modern developments". Bull. Amer. Math. Soc. 50 (4): 527–628. arXiv:1303.1856. Bibcode:2013arXiv1303.1856L. doi:10.1090/S0273-0979-2013-01423-X. S2CID 119612431.
2. Weisstein, Eric W. "Golomb-Dickman Constant Continued Fraction". mathworld.wolfram.com. Retrieved 2024-10-11.
3. Purdon, P.; Williams, J.H (1968). "Cycle length in a random function". Trans. Amer. Math. Soc. 133 (2): 547–551. doi:10.1090/S0002-9947-1968-0228032-3.
4. Weisstein, Eric W. "Golomb-Dickman Constant". mathworld.wolfram.com. Retrieved 2024-10-11.