A notable multiplicative function is the Möbius function *μ*(*n*) defined as follows:

*μ*(*n*)=1 when n is a square-free integer with an even number of distinct prime factors

*μ*(*n*)=-1 when n is a square-free integer with an odd number of distinct prime factors

*μ*(*n*)=0 when n is not square-free.

Note that *μ*(1)=1, as 1 has an even number of prime factors; namely, zero.

In terms of the distinct prime factor counting function *ω*(*n*) (see here), we can write the Möbius function as:

One can easily confirm that *μ*(*n*) is a multiplicative function (but not a completely multiplicative function); if either *m* or *n* is not square-free (or both), then *mn* is not square-free [and *μ*(*mn*)=*μ*(*m*)*μ*(*n*)=0]. If *m* and *n* are both square-free and coprime, they have no common prime factors, so *mn* is square-free and *ω*(*mn*)=*ω*(*m*)+*ω*(*n*) [so ].

In terms of the powers of primes, we see:

.

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Tags: Arithmetic Function, Math, Möbius Function, Monday Math, Multiplicative Function, Number Theory, Prime Factors

This entry was posted on August 24, 2009 at 12:07 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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