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The classical M鯾ius function is an important multiplicative function in number theory and combinatorics. It is named in honor of the German mathematician August Ferdinand M鯾ius, who first introduced it in 1831. This classical M鯾ius function is a special case of a more general object in combinatorics.
- 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.
This is taken to imply that μ(1) = 1. The value of μ(0) is generally left undefined, but the Maple computer algebra system for example returns −1 for this value.
Properties and applications
(A consequence of the fact that every non-empty finite set has just as many subsets with an even number of elements as it has subsets with an odd number of elements.) This leads to the important M鯾ius inversion formula and is the main reason that μ is of relevance in the theory of multiplicative and arithmetic functions.
Other applications of μ(n) in combinatorics are connected with the use of the Polya theorem in combinatorial groups and combinatorial enumerations.
for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta function. See the article on the Mertens conjecture for more information about the connection between M(n) and the Riemann hypothesis.
If n is a sphenic number (i.e. a product of three distinct primes), then clearly μ(n) = −1.
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 n is prime, then μ(n) = −1, but the converse is not true. The first non prime n for which μ(n) = −1 is 30 = 2. The first such numbers with 3 distinct prime factors (sphenic numbers)are ():
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 ():
2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, ...
In combinatorics, every locally finite poset is assigned an incidence algebra. One distinguished member of this algebra is that poset's "M鯾ius function". The classical M鯾ius function treated in this article is essentially equal to the M鯾ius function of the set of all positive integers partially ordered by divisibility. See the article on incidence algebras for the precise definition and several examples of these general M鯾ius functions.
- Ed Pegg's Maths Games: The M鯾ius function (and squarefree numbers)
- MathWorld: M鯾ius 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
- Sloane's On-Line Encyclopedia of Integer Sequences
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