What does $\mu^{2}$ mean?

Question

Formula 5 in The Riemann Hypothesis; Borwein, Choi, Rooney, Weirathmueller, page 74,

$$\frac{\zeta(s)}{\zeta(2s)} = \sum_{n=1}^{\infty} \frac{\mu^{2}(n)}{n^{s}},$$ $\Re(s)>1,$ where $\mu$ is the Moebius function.

What does $\mu^{2}(\cdot)$ mean?

Answer 1

$\mu^2 (n)$ denotes the squared Moebius function. The typical Moebius function returns values of $-1, 0, 1$; the squared Moebius function is obtained by simply squaring this result. So, we can break it down like this:

$\mu^2(n) = 1$ if $n$ is square free.

$\mu^2(n) = 0$ if $n$ is not.