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The Möbius function is 0 when n has a squared prime factor, and otherwise (−1)^k where k is the number of distinct primes. It is the key to Möbius inversion and appears throughout analytic number theory.
Möbius function
μ(1) = 1; μ(n) = (−1)^k if n is a product of k distinct primes; μ(n) = 0 otherwise
30 = 2 × 3 × 5 has three distinct primes and no repeats, so μ = −1.
It claimed |M(x)| < √x always. Odlyzko and te Riele disproved it in 1985, though no explicit counterexample is known.