Prime Omega Function

The prime omega function is the arithmetic function

(1)

(2)

Thus counts prime factors with multiplicity, and . It is implemented in the Wolfram Language as PrimeOmega[n].

The first few values of for , 2, ... are 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, ... (OEIS A001222). For example, because . In contrast, the prime nu function gives because it ignores multiplicity.

Conway et al. (2008) called the "multiprimality" of , and accordingly used the names biprime for a semiprime, triprime when , and so on.

The prime omega function is completely additive,

(3)

for all positive integers and . It determines the Liouville function by

(4)

Let

(5)

An asymptotic series for this mean is

(6)

where is a constant related to the Mertens constant and the are Stieltjes constants (Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003). The corresponding variance has expansion

(7)

where

			(8)
			(9)

(OEIS A091589), and

(10)

(OEIS A086242; Finch 2003) is a convergent prime sum. The coefficients are

					(11)
					(12)
					(13)
					(14)
					(15)

Equivalently, the sums occurring here have values

			(16)
			(17)

(Finch 2003).

If is chosen uniformly from 1 through , then the probability that

(18)

approaches

(19)

as (Erdős and Kac 1940; Knuth 1998, p. 384). This is the Erdős-Kac theorem for .

The average order of is (Hardy 1999, p. 51). More precisely,

(20)

where is asymptotic notation (Hardy and Ramanujan 1917; Hardy and Wright 1979, p. 355; Hardy 1999, p. 57).

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References

Conway, J. H.; Dietrich, H.; and O'Brien, E. A. "Counting Groups: Gnus, Moas, and Other Exotica." Math. Intell. 30, 6-18, 2008.Diaconis, P. "Asymptotic Expansions for the Mean and Variance of the Number of Prime Factors of a Number

." Dept. Statistics Tech. Report 96, Stanford, CA: Stanford University, 1976.Diaconis, P. "G. H. Hardy and Probability???" Bull. London Math. Soc. 34, 385-402, 2002.Erdős, P. and Kac, M. "The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions." Amer. J. Math. 62, 738-742, 1940.Finch, S. "Two Asymptotic Series." Dec. 10, 2003. https://web.archive.org/web/20160419150604/http://www.people.fas.harvard.edu/~sfinch/csolve/asym.pdf.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hardy, G. H. and Ramanujan, S. "The Normal Number of Prime Factors of a Number

." Quart. J. Math. 48, 76-92, 1917.Hardy, G. H. and Wright, E. M. §§22.10-22.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 354-355, 1979.Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 384, 1998.Knuth, D. E. Selected Papers on Analysis of Algorithms. Stanford, CA: CSLI Publications, pp. 338-339, 2000.Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. 327, 2000.Sloane, N. J. A. Sequences A001222/M0094, A086242, and A091589 in "The On-Line Encyclopedia of Integer Sequences."Turán, P. "On a Theorem of Hardy and Ramanujan." J. London Math. Soc. 9, 274-276, 1934.Turán, P. "Über einige Verallgemeinerungen eines Satzes von Hardy und Ramanujan." J. London Math. Soc. 11, 125-133, 1936.

Cite this as:

Weisstein, Eric W. "Prime Omega Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrimeOmegaFunction.html

Prime Omega Function

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