Prime Distribution

Def Prime-counting Function
We define the prime counting function $\pi (x)$ to be the number of prime numbers that less than or equal to $x$.

Thrm $\pi (x)\in \Theta (\frac{x}{\log x})$

Def $\mathrm{Li}$ Function
We define a function, $\mathrm{Li}(x)={\int }_2^x\frac{\text{dd}t}{\log t}$
Prop $\mathrm{Li}(x)>\pi (x)$ for $2\le x\le 10^{19}$, and $\mathrm{Li}(x)<\pi (x)$ for some $x\le 10^{137}$