Question

The Prime Number Theorem. ...​

Answer 1

The Prime Number Theorem States an Asymptotic relationship between number of primes less than or equal to a given number and also the Asymptotic general term for prime numbers.

i.e.,

$\pi(x) \sim \sf \frac{x}{ ln(x)}$

OR

$p_{n} \sim \sf n \: ln(x)$

We can Improvise, the Prime Number Theorem as:

$\\ \small\pi(x) \sim \sf li(x)$

where, li(x) is the logarithmic integral.