The classical Möbius function μ(n) is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832.^{[1]}^{[2]} This classical Möbius function is a special case of a more general object in combinatorics (see below).
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Definition
μ(n) is defined for all positive integers n and has its values in {−1, 0, 1} depending on the factorization of n into prime factors. It is defined as follows:
 μ(n) = 1 if n is a squarefree positive integer with an even number of prime factors.
 μ(n) = −1 if n is a squarefree positive integer with an odd number of prime factors.
 μ(n) = 0 if n is not squarefree.
An equivalent way to state this is to define the two functions
ω(n), the number of distinct primes dividing the number n and
Ω(n), the number of prime factors of n, counted with multiplicities. Clearly, ω(n) ≤ Ω(n).
Then
This implies that μ(1) = 1. (1 has an even number of prime factors, namely zero). The value of μ(0) is undefined.
Values of μ(n) for the first 25 positive numbers (sequence A008683 in OEIS):
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:
(A consequence of the fact that every nonempty 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 why μ is of relevance in the theory of multiplicative and arithmetic functions.
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