Möbius Function

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For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity. It has values in {−1, 0, 1} 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.

Euler's Totient Function

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In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n.In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1.