Computing Möbius function of the natural numbers set

Question

Let $N$ be the set of all natural numbers. Consider the poset $(N,\leq)$ where $\forall$ $r,s$ in $N$, $r\leq s$ iff $s$=$r^n$ for some $n$ in $N$. Compute the Möbius function of that poset. I have been trying to compute it, but I got nowhere. I think The Hall's Theorem may help How to calculate the mobius function of a Poset using Hall's theorem , but I have difficulty to compute the number of m-chains between $r$ and $s$. Is Hall's theorem the right way to do this problem? Any help would be appreciated.

Answer 1

The elements of a chain from $r$ to $s=r^n$ must be powers of $r$ with each exponent dividing the next one. Thus these chains are in bijective correspondence with the chains between $1$ and $n$ in the divisibility poset. Thus $$ \mu_{(\mathbb N,\le)}(r,r^n)=\mu_{(\mathbb N,\,\mid\,)}(1,n)=\mu(n)=\begin{cases} +1&\text{$n$ is squarefree with an even number of prime factors,}\\ -1&\text{$n$ is squarefree with an odd number of prime factors,}\\ 0&\text{$n$ is not squarefree.} \end{cases} $$