We know that the number $N(n,q)$ of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_q$ is given by Gauss’s formula $$N(n,q)=\frac{q-1}{n} \sum_{d\mid n}\mu(n/d)q^d.$$
The number $M(n,q)$ of irreducible polynomials of degree at most $n$ over the finite field $\mathbb{F}_q$ should be $M(n,q)= \sum_{k=1}^n N(k,q)$. Is there a nicer (shorter) formula for $M(n,q)$ as well as a sharp upper bound on it?
