## Golomb-Dickman Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let be a Permutation of elements, and let be the number of Cycles of length in this Permutation. Picking at Random gives

 (1)

 (2)

 (3)

(Shepp and Lloyd 1966, Wilf 1990). Goncharov (1942) showed that
 (4)

which is a Poisson Distribution, and
 (5)

which is a Normal Distribution, is the Euler-Mascheroni Constant, and is the Normal Distribution Function. Let
 (6) (7)

Golomb (1959) derived
 (8)

which is known as the Golomb Constant or Golomb-Dickman constant. Knuth (1981) asked for the constants and such that
 (9)

and Gourdon (1996) showed that

 (10)

where
 (11)

can be expressed in terms of the function defined by for and
 (12)

for , by
 (13)

Shepp and Lloyd (1966) derived
 (14)

Mitchell (1968) computed to 53 decimal places.

Surprisingly enough, there is a connection between and Prime Factorization (Knuth and Pardo 1976, Knuth 1981, pp. 367-368, 395, and 611). Dickman (1930) investigated the probability that the largest Prime Factor of a random Integer between 1 and satisfies for . He found that

 (15)

Dickman then found the average value of such that , obtaining
 (16)

which is .

References

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/golomb/golomb.html

Gourdon, X. 1996. http://www.mathsoft.com/asolve/constant/golomb/gourdon.html.

Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 2nd ed. Reading, MA: Addison-Wesley, 1973.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: Addison-Wesley, 1981.

Knuth, D. E. and Pardo, L. T. Analysis of a Simple Factorization Algorithm.'' Theor. Comput. Sci. 3, 321-348, 1976.

Mitchell, W. C. An Evaluation of Golomb's Constant.'' Math. Comput. 22, 411-415, 1968.

Purdom, P. W. and Williams, J. H. Cycle Length in a Random Function.'' Trans. Amer. Math. Soc. 133, 547-551, 1968.

Shepp, L. A. and Lloyd, S. P. Ordered Cycle Lengths in Random Permutation.'' Trans. Amer. Math. Soc. 121, 350-557, 1966.

Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1993.