irreducible monic polynomials

Question

Let $P_1,P_2,\dots $ be the irreducible monic polynomials in $\mathbb{F}_p[x]$. Is there any possibility to prove the following

$$\lim_{n\to \infty } \sum_{i_1+\cdots+i_n=t}z^{i_1\deg P_1 + \dots +i_n\deg P_n} = (pz)^t $$ for $t\in \mathbb{N}$ and $\vert z \vert < 1/p$?

Answer 1

In a comment you write

I have to prove: $$\prod_{r=1}^{\infty }(1+z^{\deg P_r}+z^{2\deg P_r}+\cdots ) = 1+pz+p^2z^2+p^3z^3+\cdots$$

Since there are only finitely many polynomials of degree $\leqslant n$ in $\mathbb{F}_p[x]$, for every fixed power of $z$, only finitely many factors of the infinite product are relevant, hence we need not occupy ourselves with convergence issues.

Now, the coefficient of $z^t$ on the right hand side is $p^t$, which is the number of monic polynomials in $\mathbb{F}_p[x]$. On the left hand side, the coefficient of $z^t$ is the number of ways of writing $t$ in the form

$$\sum_{k=1}^n i_k\cdot \deg P_k$$

where $n$ is chosen so that $\deg P_r > t$ for all $k > n$, and the $i_k$ are integers. Note that

$$\sum_{k=1}^n i_k\deg P_k = \deg \prod_{k=1}^n P_k^{i_k},$$

so the coefficient of $z^t$ on the left hand side counts the number of factorisations of monic polynomials of degree $t$ into powers of the $P_k$. Since $\mathbb{F}_p[x]$ is a unique factorisation domain, we have a bijection between the monic polynomials of degree $t$ and the factorisations, hence the coefficients are the same.