Golomb–Dickman constant
In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is
- [math]\displaystyle{ \lambda = 0.62432 99885 43550 87099 29363 83100 83724\dots }[/math] (sequence A084945 in the OEIS)
It is not known whether this constant is rational or irrational.[1]
Definitions
Let an 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
- [math]\displaystyle{ \lambda = \lim_{n\to\infty} \frac{a_n}{n}. }[/math]
In the language of probability theory, [math]\displaystyle{ \lambda n }[/math] 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,
- [math]\displaystyle{ \lambda = \lim_{n\to\infty} \frac1n \sum_{k=2}^n \frac{\log(P_1(k))}{\log(k)}, }[/math]
where [math]\displaystyle{ P_1(k) }[/math] is the largest prime factor of k (sequence A006530 in the OEIS) . So if k is a d digit integer, then [math]\displaystyle{ \lambda d }[/math] 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 [math]\displaystyle{ \lambda }[/math]. More precisely,
- [math]\displaystyle{ \lambda = \lim_{n\to\infty} \text{Prob}\left\{P_2(n) \le \sqrt{P_1(n)}\right\} }[/math]
where [math]\displaystyle{ P_2(n) }[/math] 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 [math]\displaystyle{ f^{n+k}(x) = f^n(x) }[/math] for sufficiently large n; the smallest k with this property is the length of the cycle. Let bn 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[2] proved that
- [math]\displaystyle{ \lim_{n\to\infty} \frac{b_n}{\sqrt{n}} = \sqrt{\frac{\pi}{2} } \lambda. }[/math]
Formulae
There are several expressions for [math]\displaystyle{ \lambda }[/math]. These include:
- [math]\displaystyle{ \lambda = \int_0^1 e^{\mathrm{Li}(t)} \, dt }[/math]
where [math]\displaystyle{ \mathrm{Li}(t) }[/math] is the logarithmic integral,
- [math]\displaystyle{ \lambda = \int_0^\infty e^{-t - E_1(t)} \, dt }[/math]
where [math]\displaystyle{ E_1(t) }[/math] is the exponential integral, and
- [math]\displaystyle{ \lambda = \int_0^\infty \frac{\rho(t)}{t+2} \, dt }[/math]
and
- [math]\displaystyle{ \lambda = \int_0^\infty \frac{\rho(t)}{(t+1)^2} \, dt }[/math]
where [math]\displaystyle{ \rho(t) }[/math] is the Dickman function.
See also
External links
- Weisstein, Eric W.. "Golomb-Dickman Constant". http://mathworld.wolfram.com/Golomb-DickmanConstant.html.
- 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. https://archive.org/details/mathematicalcons0000finc.
References
- ↑ Lagarias, Jeffrey (2013). "Euler's constant: Euler's work and modern developments". Bull. Amer. Math. Soc. 50 (4): 527–628. doi:10.1090/S0273-0979-2013-01423-X. Bibcode: 2013arXiv1303.1856L.
- ↑ 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.
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