A question related to study of functions in Prime polynomial theorem

Question

I am studying prime polynomial theorem from notes of a friend and I am unable to prove a lemma.

Let $F_{q}[t]$ be the ring of polynomials with coefficients in $F_q$ . Also, let $\pi_{q}(n) $ be the number of monic irreducible $P \in F_q[x] $ of degree $n$.

Let von Mangoldt function be defined as $\Lambda (f) =\deg P$ , if $ f= cP^k $ is a power of a prime $P$ and $0$ otherwise.

Also, define $\psi(n) = \sum_{\substack{\deg f=n\\f\text{ monic}}} \Lambda(f) $.

Prove that $\psi(n) = \sum_{d\mid n} d \pi_{q} (d) $ .

Only $f = P^k$ case need to be dealt due to nature of Mangoldt function.

LHS :$\psi(n) = \sum_{\substack{\deg f =n\\f\text{ monic}}} \deg P = \sum_{\deg f=n} n/k$ but I am not able to simplify after it.

RHS: $\sum_{d| n} d \pi_{q} (d) = \sum_{d|n} d$ .

I am not able to proceed further despite thinking a lot.

Can you please help.

Answer 1

I think you are perhaps overthinking it. \begin{equation} \psi (n) = \sum_{\deg f = n, f \text{ monic}} \Lambda(f) = \sum_{\deg f = n, f = P^k, P \text{ monic irreducible}} \deg(P) \end{equation} Thus \begin{equation} \psi (n) = \sum_{P \text{ monic irreducible}, \deg(P)|n} \deg(P) = \sum_{d|n} \# \{ P \text{ monic irreducible} | \deg(P) = d\} * d = \sum_{d|n} d \pi_q(d) \end{equation}