Möbius Function

The function \( μ(n) \), called the Möbius function, is defined for any integer \( n> 0 \) of the set \( \mathbb{N}* \) in the set of 3 values \( \{-1, 0, 1 \} \).

\( μ(n) \) is \( 0 \) if \( n \) has for divisor a perfect square (other than 1)

\( μ(n) \) is \( 1 \) if \( n \) has for divisors an even number of prime numbers

\( μ(n) \) is \( -1 \) if \( n \) has for divisors an odd number of prime numbers

The image of \( μ(n) \) depends on the prime number decomposition of \( n \). If a prime number appears several times in the decomposition, then \( μ(n) = 0 \), otherwise, if the decomposition has an even number of prime numbers, then \( μ(n) = 1 \) and otherwise with an odd number of prime numbers \( μ(n) = -1 \).

Example: \( 12 = 2 \times 2 \times 3 \) so \( μ(12) = 0 \) because \( 2 \) appears twice, and so \( 12 \) is divisible by \( 4 \), a perfect square

Example: \( 1234 = 2 \times 617 \) therefore \( μ(12) = 1 \) because the decomposition has 2 distinct primes (2 is an even number)

Example: \( 12345 = 3 \times 5 \times 823 \) so \( μ(12) = -1 \) because the decomposition has 3 distinct prime numbers (3 is an odd number)