Möbius function

The classic Möbius function μ(n) 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.

Definition

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.

This is taken to imply that μ(1) = 1. For n= 0 μ(n) is generally left undefined; the Maple computer algebra system for example returns -1 for this value.

Maple calling sequence notation:

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

or:

 > numtheory[mobius](n);

Properties and Applications

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:

<math>M(n) = \sum_{1\le k \le n} \mu(k)</math>

for every natural number n. This function is closely linked with the positions of zeroes of the Euler - Riemann ζ- function. The connection between M(n) and the Riemann conjecture was known to Thomas Joannes Stieltjes. See the article on the Mertens conjecture[?] for more information about this connection.

μ(n) sections

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 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*3*5. The first such numbers with 3 distinct prime factors are (SIDN A007304):

  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(n) = 0, for example μ(4620) = 0, as 4620 = 2 ² * 3 * 5 * 7 * 11 (SIDN A051270):

  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,...

See also:

• Eric Weissten's definition and some properties of μ(n): http://mathworld.wolfram.com/MoebiusFunction
• Some futher 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