Möbius function

The Möbius function assigns zero to integers divisible by a squared prime and otherwise assigns [0m[0m or [0m[0m according to the parity of the number of distinct prime factors. It is central to Möbius inversion in number theory.

Square-free integer: An integer not divisible by the square of any prime. Exactly the positive integers on which the function is nonzero.
Euler's totient function: The function counting positive integers up to n that are coprime to n. A divisor sum involving the Möbius function gives an exact formula for it.
August Ferdinand Möbius: A German mathematician and astronomer who introduced the function now bearing his name. His 1832 work established the function and its inversion formula.
Liouville function: An arithmetic function equal to negative one raised to the total number of prime factors, counted with multiplicity. Unlike the Möbius function, it never vanishes on positive integers.
Prime factorization: A representation of an integer as a product of primes with multiplicities. Distinct prime factors determine the function's sign; repeated factors force zero.
Primitive necklace: A circular arrangement of colored beads that is not a repetition of a shorter arrangement. Möbius inversion counts necklaces with no smaller repeating period.
Möbius inversion: A general method for reversing sums over partially ordered sets, including integer divisors. The arithmetic function's name and classical role come from this inversion principle.
Mangoldt function: An arithmetic function equal to log p on prime powers and zero otherwise. It singles out prime powers directly instead of encoding square-free divisors.
Divisor function: An arithmetic function that counts or sums over the positive divisors of an integer. The Möbius function's divisor sums produce identities used in inversion.
Primitive word: A finite word that is not a power of a shorter word. Inversion separates primitive words from words formed by repetition.

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