Can you explain the following property for arithmetic functions.

Question

I have trouble understanding the following identity $$\prod_{d|2n} (x^d-1)^{\mu (2n/d)} = \prod_{d|n} (x^d-1)^{\mu (2n/d)} \prod_{d|n} (x^{2d}-1)^{\mu ((2n/d)/2)}$$, $\mu (n)$ is the Mobius function. Can you explain this? Also can you explain how to turn expressions of the form $\prod_{d|f(n)} g(n,d)$ into expressions involving $\prod_{d|n}$ for some functions $f,g$ and integers $n,d$. Or at least for $f(n)$ of the form $f(n)=an+b$ for some integers $a,b,n$

Answer 1

Consider the cases where $d$ is even and odd separately. If $d$ is odd then $d|2n\iff d|n$ so the product over odd factors is the first term on the right-hand side.

If $d$ is even, say $d=2d',$ then $d|2n \iff d'|n$ and the second term on the right is the product over the $d'$ with $d$ written in place of $d'$.