The Prime Number Theorem - Revisited

The Prime Number Theorem - Revisited

The Prime Number Theorem is one of the most famous theorems in mathematics. It tells us something about the distribution of the prime numbers.

How many of the first n integers 1,2,3,4,....,n are prime? The Prime Number Theorem says the number of primes is approximately n/log(n)

This is not an exact count, n/log(n) is only an approximation, but as n gets bigger the approximation gets better and better.

The Prime Number Theorem is also a statement about the information entropy of the primes! Here's how..

Suppose you have a machine with a big red button. Each time you punch the button the machine responds by outputting an integer in the range 1,2,3,....,n. After much experimentation you discover that the probability of getting integer j is pj. Then physics defines the information entropy of this machine as..

information entropy=(-1)*sum (pj*log(pj)) for j=1,2,3,...,n

In the special case where all numbers occur with equal probability pj=1/n for all j and we get the famous result for the entropy of the machine..


Now imagine this entropy is "distributed" equally across all numbers, so on average an individual integer has log(n)/n entropy.

If the integers 1,2,3,....,n contain m primes then the entropy of the primes is simply m*log(n)/n

But the prime number theorem says that m=n/log(n) approximately. So the approximate entropy becomes..


and as n approaches infinity this approximation becomes exact. So we can say that..

"The information entropy of the primes is 1".

This statement is equivalent to the prime number theorem. How strange!

Content written and posted by Ken Abbott