Monday Math 86

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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2 Responses to “Monday Math 86”

  1. Monday Math 91 « Twisted One 151’s Weblog Says:

    […] of multiplicative functions much easier. For example, consider the Dirichlet convolution of the Möbius function μ(n), and the multiplicative function 2ω(n) appearing in this post. Since both are […]

  2. Monday Math 88 « Twisted One 151's Weblog Says:

    […] a product over primes: . Now, let us use this for a couple of multiplicative functions. First, the Möbius function μ(n). We recall that Thus, the series inside the prime product has only two non-zero terms, […]

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