Is $\pi(n)$ a Rational Function? Are there some two-variable polynomials $P(n,\log n)$ and $Q(n,\log n)$ which we have the bellow equation for prime counting function $\pi(n)$ for $n \in \mathbb{n}$?
$$\pi(n) =  \Bigl{\lfloor} \frac{P(n,\log n)}{Q(n, \log n)} \Bigr{\rfloor}$$
Note: According to the Prime Number Theorem such these polynomials should satisfy the bellow limit condition.
$$\lim_{n\to\infty} \frac{P(n,\log n) \times \log n}{Q(n, \log n) \times n}  =1 $$
 A: With looking to the method advised in the Jeppe Stig Nielsen's comment we can complete the proof for the claim which there aren't such these polynomials.
In this paper it's proved which for every $m$ we have:
$$\liminf_{n\to \infty} p_{n+m} - p_{n} < \infty $$
and thus if the above polynomials $P(x)$ and $Q(x)$ exist then for some fixed $k$ and infinitely many $n$s we should have
$$ \frac{P(n+m,\log (n+m))}{Q(n+m, \log (n+m))} - \frac{P(n,\log n)}{Q(n, \log n)} > k-1 $$
and for some other $n$s (again, infinitely many times) because of existance of intervals of length $m+2$ which are free of the primes(haven't any primes) we should have
$$ \frac{P(n+m,\log (n+m))}{Q(n+m, \log (n+m))} - \frac{P(n,\log n)}{Q(n, \log n)} <1 $$
this means which the rational function $ \frac{P(n+m,\log (n+m))}{Q(n+m, \log (n+m))} - \frac{P(n,\log n)}{Q(n, \log n)}  $ which is a function of $n$ should oscillate infinitely many times between $1$ and $k-1$ but such a function for large $n$'s is increasing or decreasing(this claim can be proved just with calculating the derivative) and thus $\pi(n)$ can not be written as a rational function of the form $\Bigl{\lfloor} \frac{P(n,\log n)}{Q(n, \log n)} \Bigr{\rfloor}$.
