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# Prime Number Theorem

When prime numbers are considered, we know that there are infinite number of them. To understand this concept of infinite number here, we use a theorem called the prime number theorem.
The prime number theorem defines about the density of prime numbers that is how prime numbers are distributed among all the positive integers.

Let assume $n$ is any positive real number and let $\pi(x)$ denotes the number of primes numbers less than or equal to x.
So  ${\pi (n)\to \propto }$,  as ${(n)\to \propto }$
For example $\pi(30)$ denotes  the number of prime numbers less than 30.
The prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
So $\pi(30)$  =  10.
For small number, its possible to estimate all prime number, but as the numbers get larger, it can be seen that these prime numbers are irregularly distributed. In such situation prime number theorem helps a lot to get an approximate answer.

The prime number theorem implies that the probability that a random number n, which is selected from a set of positive number, will be prime, is about $\frac{\ (1)}{ln(x)}$.
This means in a group of n numbers, $1$ out of every $ln(n)$ will be a prime.
For example among the positive integers up to and including n = 153, about 1 in $ln(153)=5.03\sim 5$ will be a prime number.

So the prime number theorem states that, the limit of the quotient of two functions $\pi(n)$ and $\frac{\ (n)}{ln(n)}$ will be $1$ as n tends to infinity.
The asymptotic law of distribution of prime numbers, can be formulated as
$\lim_{x\to \propto}\frac{\pi(x)}{x/ln(x)}= 1$
Hence, prime number theorem can be restated as $\pi(n)\sim\frac{\ (n)}{ln(n)}$
This means that $\frac{\ (n)}{ln(n)}$is a good approximation for $\pi(n)$
It could be restated in another way also as
$\pi {(n)\sim \int_{0}^{x}\frac{dt}{ln(t)} }$

This was discovered by Gauss and so its also called Gauss prime number theoem.

## Chebyshev Prime Number Theorem

$\theta (x)=\sum_{p\leqslant x}ln(p)$  and $\psi (x)=\sum_{p^{m}\leqslant x}ln(p)$
$\lim_{x\to \propto }\frac{\theta (x)}{x}=1$