The following is from this paper that discusses polynomials and classic number theory functions. Theorem:
There do not exist polynomials $P,Q \in \mathbb{R}[X]$ such that $\displaystyle\int_{0}^{\log n}{\frac{P(x)}{Q(x)}dx} = \frac{n}{\pi(n)},$ for all positive integers $n \geq 1$.
Proof: Suppose such polynomials exist. Let R be that rational funciton and put $$f(x) \frac{1}{x}\int_{0}^{\log x}R(t)dt.$$ Then we have $f(n) = f(n+1)$ whenever $n+1$ is composite. Thus, $f'$ vanishes infinitely many times by Rolle's Theorem, so there is a sequence $c_n$ between n and $n+1$ whenever $n+1$ is composite such that $f'(c_n) = 0$. Since, $$f(x) + xf'(x) = \frac{1}{x}R(\log x),$$ this gives $$R(\log(c_n)) = \int_{0}^{\log(c_n)}R(t)dt,$$ which means that by asymptotic considerations that R must be null, contradiction.
What does he mean "asymptotic considerations"? I can't see what contradiction he's referring to that arises out of that last equality.