Square-free integer

A square-free integer is a number which is not divisible by any square numbers other than 1. In other words, each prime number that appears in its prime factorization appears exactly once.

For example, ${\displaystyle 6=2\times 3}$ is square-free. However, ${\displaystyle 27=3^{3}$ is not square-free: it is divisible by ${\displaystyle 9=3^{2}$, and the power of ${\displaystyle 3}$ in the prime factorization is to a power larger than one.

The Möbius function is a function which takes in natural numbers and is usually written as ${\displaystyle \mu (n)}$. The value of ${\displaystyle \mu (n)}$ depends on whether or not ${\displaystyle n}$ is square-free. Specifically,

${\displaystyle \mu (n)={\begin{cases}1,&{\text{if }n{\text{ is square-free and has an even number of prime factors},\\-1,&{\text{if }n{\text{ is square-free and has an odd number of prime factors}\\0,&{\text{if }n{\text{ is not square-free}\end{cases}$

For example, ${\displaystyle 6=2\times 3}$ is square-free with an 2 prime factors, so ${\displaystyle \mu (6)=1}$. Since ${\displaystyle 27=3^{3}$ is not square-free, then ${\displaystyle \mu (27)=0}$.

Since ${\displaystyle 5}$ is prime, it is its own prime decomposition. That is, the prime factorization of ${\displaystyle 5}$ has 1 prime factor, so ${\displaystyle \mu (5)=-1}$.

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