In Tom Apostol's Analytic Number Theory book there is a problem which states:
- That there do not exists polynomials $P(x)$ and $Q(x)$ such that $\pi(x) = \frac{P(x)}{Q(x)}$ for all $x \in \mathbb{N}$.
I tried this problem, but couldn't find a solution. Here is what i attempted. Since $\pi(x) \sim x\log{x}$ as $x \to \infty$, i saw what happens to the Right hand side $\frac{P(x)}{Q(x)}$ as $x \to \infty$. I couldn't conclude anything. If there are interesting proof's for this result, i shall be happy to see it.