The Möbius Function

Number Theory · Axiom Academy

A fundamental multiplicative function in number theory with deep connections to prime factorization 1. Definition of the Möbius Function The Möbius function μ(n) is defined for positive integers n as: μ(n) = 0 if n has a squared prime factor (not square-free) μ(n) = (-1) k if n is the product of k distinct primes 2. Computing Möbius Function Values Let's compute μ(n) for the first several positive integers by examining their prime factorizations: n = 1: μ(1) = 1 (by definition) n = 2: 2 is prime → 1 distinct prime → μ(2) = (-1) 1 = -1 n = 3: 3 is prime → 1 distinct prime → μ(3) = -1 n = 4: 4 = 2² → has squared factor → μ(4) = 0 n = 6: 6 = 2 × 3 → 2 distinct primes → μ(6) = (-1) 2 = 1 n = 12: 12 = 2² × 3 → has squared factor → μ(12) = 0 n = 30: 30 = 2 × 3 × 5 → 3 distinct primes → μ(30) = (-1) 3 = -1 The Möbius function has several remarkable properties: Multiplicativity: For coprime integers m and n (gcd(m,n) = 1): Summation Property: The sum of μ(d) over all divisors of n equals: This identity is fundamental to the Möbius inversion formula. 4. Applications and Significance The Möbius function appears throughout number theory: If f and g are arithmetic functions related by: Then we can invert this to get: The totient function φ(n) can be expressed using μ:

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