A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb Q_+\!\!\rightarrow\mathbb Q_+$:
$\psi(a\cdot b)=\psi(a)\cdot\psi(b)\Rightarrow\psi(\prod p_i^{n_i})=\prod\psi(p_i)^{n_i}$, so any function $i\mapsto \psi(p_i)$ defines $\psi$. Further, since $\varphi(\frac{1}{n})=\frac{1}{\varphi(n)}$ a function $f$ uniquely defines the group homomorphism $\varphi$.
For fun I intend to study the group homomorphism generated by the identity function $i\mapsto\omega(p_i)=i$. Then $\omega:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ seems to be an algebraic analogue to the analytic approach with $\pi(N)$ since also $\omega$ is a left inverse to the prime number function $p_n$.
In the diagram $\omega$ is the blue curve and $\pi$ the red and one can see (especially if the picture is loaded and viewed in a larger scale) that $\omega(1)>\pi(1),\; \omega(35)>\pi(35)$ and $\omega(49)>\pi(49)$. Of course $\omega$ and $\pi$ coincide for all primes, but also for other values as $91,\;95$ and $133$. However, I have tested all values less than $2^{16}$ and $\omega(N)\leq\pi(N)$ for $N>49$ in this interval.
I have no idea how to prove the conjecture (if possible) but I will try to construct a program that (in principle) can test any interval of integers.
Edit: I realize now that it is virtually impossible to make such a program, but with help of arrays of primes my program have tested all values $< 1\,000\,000$ and the only numbers for which $\omega(N)\geq\pi(N)$ (italic numbers stands for '$>$') are:
$\mathit{1},\,9,\,15,\,21,\,25,\,\mathit{35},\,39,\,\mathit{49},\,57,\,65,\,91,\,95,\,133$.
Conjecture: For $N=\prod p_k^{n_k}>49$ it holds that $\prod k^{n_k}\leq\pi(N)$.
Thoughtful: from an analytic point of view $\omega$ seems to be very irregular, but algebraically it is an almost canonical morphism.