About the Möbius Function
The Möbius function μ(n) is an important function in number theory that helps study the properties
of integers through their prime factorizations.
Function Values:
- μ(1) = 1
- μ(n) = 0 if n has a squared prime factor
- μ(n) = (-1)ᵏ if n is a product of k distinct primes
Examples:
- μ(1) = 1
- μ(2) = -1 (one prime factor)
- μ(6) = 1 (two prime factors: 2 × 3)
- μ(8) = 0 (includes 2²)
Properties:
- The function is multiplicative
- ∑μ(d) = 1 if n = 1, and 0 if n > 1 (sum over all divisors d of n)
- Used in the formulation of the Möbius inversion formula
Applications:
- Number theory
- Combinatorics
- Analytic number theory
- Inclusion-exclusion principle