The classic **Möbius function** μ(*n*), named in honor of August Ferdinand Möbius, is an important multiplicative function considered in number theory and in combinatorics. Combinatorialists assign to every locally finite poset
an incidence algebra, one member of which is that poset's "Möbius function". The classic Möbius function treated in this article is the Möbius function of the set of all positive integers partially ordered by divisibility.
The Möbius function is named for German mathematician August Ferdinand Möbius, who first introduced it in 1831.

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2 Properties and Applications 3 μ( n) sections4 External links |

μ(*n*) is defined for all positive natural numbers *n* and has its values in {-1, 0, 1} depending on the natural or integer factorization of *n*. It is defined as follows

- μ(
*n*) = 1 if*n*is a square-free positive integer with an even number of distinct prime factors. - μ(
*n*) = -1 if*n*is a square-free positive integer with an odd number of distinct prime factors. - μ(
*n*) = 0 if*n*is not square-free.

Maple calling sequence notation:

> with(numtheory): > mobius(n);or:

> numtheory[mobius](n);

The Möbius function is multiplicative and is of relevance in the theory of multiplicative and arithmetic functions because it appears in the Möbius inversion formula.

Other applications of μ(*n*) in combinatorics are connected with the use of the Polya theorem in combinatorial groups and combinatorial enumerations.

In number theory another arithmetic function closely related to the Möbius function is very important; it is defined by:

μ(*n*) = 0 if and only if *n* is divisible by a square. The first numbers with this property are (Sloane ID Number A013929 ??):

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63,...If

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222,...and the first such numbers with 5 distinct prime factors are (SIDN A046387):

2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, 9870, 10010,10230,10374,10626,11130,11310,11730,12090,12210,12390,12558,12810, 13090,13110...Very similar numbers to the above ones are (not necessarily square-free) numbers with exactly 5 different prime factors. Some of these can have μ(

2310, 2730, 3570, 3990, 4290, 4620, 4830, 5460, 5610, 6006, 6090, 6270, 6510, 6630, 6930, 7140, 7410, 7590, 7770, 7854, 7980, 8190, 8580, 8610, 8778, 8970, 9030, 9240, 9282, 9570, 9660, 9690, 9870,10010,10230,10374, 10626,10710,10920,11130,...

- Ed Pegg's Maths Games: The Möbius function (and squarefree numbers)
- MathWorld entry for the Möbius function
- Some further applications of μ(
*n*) as its physical interpretation, specifically treated as the operator (-1)^{F}what is equivalent to the*Pauli exclusion principle*: http://www.maths.ex.ac.uk/~mwatkins/zeta/wolfgas.htm