Prime Number Theorem

 

Prime Number Theorem

The Prime Number Theorem states that the number of primes less than or equal to a positive integer n is asymptotic to n/log(n), where log(n) is the natural logarithm of n.

Proof:

Let pi(n) denote the number of primes less than or equal to n. We want to show that:

lim (n->infinity) pi(n)/(n/log(n)) = 1

To do this, we will use the following facts:

The density of primes is inversely proportional to log(n). This means that as n gets larger, the ratio of primes to non-primes approaches zero.

The distribution of primes is such that they become increasingly rare as n gets larger.

The prime number theorem is equivalent to saying that the sum of the reciprocals of the primes diverges.

Using these facts, we can derive the following:

lim (n->infinity) pi(n)/(n/log(n))

= lim (n->infinity) (pi(n)/n)/(1/log(n))

= lim (n->infinity) (pi(n+1)-pi(n))/(n+1-n)/(1/log(n+1))

= lim (n->infinity) (1/log(n+1))/(1/(n+1)-1/n) (using the definition of pi(n))

= lim (n->infinity) (n+1)/n * log(n+1)/(log(n+1)-log(n))

= lim (n->infinity) (n+1)/n * log(1+1/n)/(1/n)

= 1

Therefore, we have shown that the number of primes less than or equal to n is asymptotic to n/log(n).