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Concept
Möbius function
Summary
The Möbius function μ(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted μ(x).
For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity. It has values in depending on the factorization of n into prime factors:
μ(n) = +1 if n is a square-free positive integer with an even number of prime factors.
μ(n) = −1 if n is a square-free positive integer with an odd number of prime factors.
μ(n) = 0 if n has a squared prime factor.
The Möbius function can alternatively be represented as
where δ 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.
It can also be defined as the Dirichlet convolution inverse of the constant-1 function.
The values of μ(n) for the first 50 positive numbers are
The first 50 values of the function are plotted below:
Larger values can be checked in:
Wolframalpha
the b-file of OEIS
The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have
This may be seen from its Euler product
Also:
where - Euler's constant.
The Lambert series for the Möbius function is:
which converges for < 1. For prime α ≥ 2, we also have
Gauss proved that for a prime number p the sum of its primitive roots is congruent to μ(p − 1) (mod p).
If Fq denotes the finite field of order q (where q is necessarily a prime power), then the number N of monic irreducible polynomials of degree n over Fq is given by:
The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry.
Official source
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