## The Mobius Function

The Mobius function is one of a class of arithmetic functions – complex or real valued functions defined on the positive integers.

The Mobius function is labelledand defined as

Ifthen writeThenotherwise.

Note thatif and only ifhas a square factor greater than 1.

Values offorto 10 are given in the following table.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

1 | -1 | -1 | 0 | -1 | 1 | -1 | 0 | 0 | 1 |

The most fundamental property of the Mobius function is that it gives a simple formula for the divisor sumextended over the positive divisors of

Theorem

Ifthen

Proof: The formula is clearly true ifSuppose then thatIn the sumthe only nonzero terms come fromand those divisors ofwhich are products of distinct primes. Thus

The Mobius function is not multiplicative, so does not obey the formulaTo see this putthenbut