The Frequency of Prime Numbers – The Prime Number Theorem

 

As we progress through the first twenty natural numbers, there are four prime numbers in the first ten numbers (2, 3, 5, 7) and another four in the next ten numbers (11, 13, 17, 19). So, in these first twenty, we have a relative frequency of 40% prime numbers. After this point, the percentage of prime numbers begins to drop. Under 40, there are 30% prime numbers; under 100, 25% prime. Moving to 1000, there are 168 prime numbers, so the frequency drops to 16.8%.

For centuries, this persistent drop of prime numbers has been known and is termed the Prime Number Theorem (PNT). It states that primes number distribution asymptotically drops. There is a function, called the prime-counting function and uses the expression, π(N), as the number of primes less than or equal to N.

A first approximation of this function is:

π(x) ~ x/ln(x)

A result of this function is that the nth prime number, p_n, can be approximated to:

p_n ~ n*ln(n)

The prime-counting function was also used in a prior post Math Vacation: Counting Prime Numbers (jamesmacmath.blogspot.com).