This is a question about a passage in a particular paper, Polynomial Functions (mod m) (Mullen and Stevens, 1984). The authors write:
Let $\tau(m)$ denote the number of functions on $\mathbb{Z}/m\mathbb{Z}$ which can be realized as polynomials. It is an easy consequence of the Chinese remainder theorem that $\tau(m)$ is multiplicative, so that only $\tau(p^n)$ for each prime $p$ and exponent $n$ need be evaluated.
What I gleaned from this is that if for example $m=p^kq^\ell$, $p,q$ prime, then $\tau(m)=\tau(p^k)\cdot\tau(q^\ell)$. I essentially have two related questions about this.
First, did I understand the point correctly?
And if so, why is it not the case that $\tau(p^n)=\tau(p)^n$?