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

Möbius function

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

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

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

References

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