Min number of primes up to n

Question

According to the prime number theorem there are about $n/\ln(n)$ primes less than $n$. This value is a limit but it could fluctuate. My question is, is there a known bound on this fluctuation? i.e. are there functions $f$ and $g$ such that $\forall n>100, f(n) < \mathrm{numPrimes}(n) \lt g(n)$? How tight is this bound? Could there be an $n$ for which the number of primes less than $n$ is smaller than $n/\sqrt{n}$? Could there be an $n$ for which there are more than $n/2$ (for $n\gt 100$ let's say) primes less than $n$?

Trivially $0 \lt\mathrm{numPrimes}(n) \lt n$, how tight a bound has been achieved?

Answer 1

Let $\pi(x)$ denote the prime counting function. Pierre Dusart has shown that $$\pi(x) \geq \frac{x}{\log(x)-1} \qquad \qquad \mbox{ for }x\geq 5393$$ and $$\pi(x) \leq \frac{x}{\log(x)-1.1}\qquad \qquad \mbox{ for } x\geq 60184.$$

Many other results of a similar flavor are also shown in this paper.

Answer 2

I have understood it to be an asymptotic expression, i.e. $$ \pi(x)\sim \frac{x}{\log x-1} $$ for large $x$. That is to say, for any positive functions $f(x)>0$ and $g(x)>0$ there exists an $n$ such that $$ \frac{x}{\log x-1}-f(x)<\pi(x)<\frac{x}{\log x-1}+g(x). $$ for all $x>n$.