# Bounds for prime counting function

The prime counting function $$pi(x)$$ is defined as

$$begin{equation} pi(x)=sum_{pleq x}1 end{equation}$$

where $$p$$ runs over primes.
I have seen many bounds for $$pi(x)$$ such as

$$begin{equation} frac{x}{log x}left(1+frac{1}{2log x}right)
$$begin{equation} frac{x}{log x – 1/2}
$$begin{equation} frac{x}{log x+2}

Till now, what are the best known upper and lower bounds for the prime-counting function? Is there a better bound that $$mathrm{Li}(x)$$?