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)