Möbius function

The classical Möbius function $\!\,\mu(n)$ is an important multiplicative function in number theory and combinatorics. It is named in honor of the German mathematician August Ferdinand Möbius, who first introduced it in 1831. This classical Möbius function is a special case of a more general object in combinatorics.

Definition

μ(n) is defined for all strictly positive natural numbers n and has its values in {, , } depending on the factorization of n into prime factors. 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. The value of μ(0) is generally left undefined, but the Maple computer algebra system for example returns −1 for this value.

The 50 first values of the function are plotted below

Properties and applications

The Möbius function is multiplicative (i.e. μ(ab) = μ(a) μ(b) whenever a and b are coprime). The sum over all positive divisors of n of the Möbius function is zero except when n = 1:

\sum_{d | n} \mu(d) = \left\{\begin{matrix}1&\mbox{ if } n=1\\

0&\mbox{ if } n>1\end{matrix}\right.

(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 Möbius 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.

In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by

M(n) = \sum_{k = 1}^n \mu(k)

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.

The Lambert series for the Möbius function is

\sum_{n=1}^\infty \frac{\mu(n)q^n}{1-q^n} = q.

μ(n) sections

μ(n) = 0 if and only if n is divisible by a square. The first numbers with this property are (sequence A013929 in the On-Line Encyclopedia of Integer Sequences):

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 (sphenic numbers) are (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 (A046387):

2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, ...

Generalization

In combinatorics, every locally finite poset is assigned an incidence algebra. One distinguished member of this algebra is that poset's "Möbius function". The classical Möbius function treated in this article is essentially equal to the Möbius 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 Möbius functions.

Physics

The Möbius function can be interpreted in physics, in the context of a theory with a logarithmic energy spectrum, as the operator (−1)^F that distinguishes fermions and bosons. The fact that μ(n) vanishes when n is not squarefree is equivalent to the Pauli exclusion principle. This identification allows for a supersymmetric interpretation of the Möbius inversion formula.

References

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0387901639
Ed Pegg Jr., "The Möbius function (and squarefree numbers)", MAA Online Math Games (2003)