Möbius function
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. It is a special case of a more general object in combinatorics.
For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity. It has values in {−1, 0, 1} depending on the factorization of n into prime factors:
The Möbius function can alternatively be represented as
where δω(n)Ω(n) is the Kronecker delta, λ(n) is the Liouville function, ω(n) is the number of distinct prime divisors of n, and Ω(n) is the number of prime factors of n, counted with multiplicity.
The values of μ(n) for the first 30 positive numbers (sequence in the OEIS) are
The first 50 values of the function are plotted below:
The Möbius function is multiplicative (i.e. μ(ab) = μ(a) μ(b) whenever a and b are coprime). The sum of the Möbius function over all positive divisors of n (including n itself and 1) is zero except when n = 1:
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