Let $q$ be some prime number. Define, for $T<|\frac{1}{q}|$, $Z(T)=\sum_{f\in M_q}T^{deg(f)}$, where $M_q$ is the set of monic polynomials in $\mathbb{F}_q[x]$. Prove that $Z(T)=\prod_{p}\frac{1}{1-T^{deg(p)}}$, where $p$ are the irreducible monic polynomials in $\mathbb{F}_q[x]$.
This seems to call for a combinatoric proof but I can't wrap my head around this. Would appreciate any hints!
By unique factorization $$\prod_{f \in F_q[x] \ monic \ irreducible}(1+\sum_{k\ge 1} T^{\deg(f^k)})=\sum_{g\in F_q[x]\ monic} T^{\deg(g)}$$ Can you finish ?