The number of primes less than or equal to a number n is defined as p(n) and its product with its natural Napierian logarithm divided by n approaches unity as n approaches infinity. This was conjectured by both Gauss and Legendre and also indirectly by Euler in the middle of the 19th century. Since they all used the analytic theory of limits and the continuity theory of the calculus, p(n) can only give approximately the discrete counting value of itself. This approximate value asserts that for very large n, about 1 in log(n) of n is a prime number. For example, if n is one million then 1 in every 14 numbers is a prime. If n=10^50 then about 1 in 115 numbers is a prime.

Then 4 years before the close of the 19th century, two mathematicians namely: J. Hadamard and C. J. de la Vallée-Poussin proved independently Euler-Gauss-Legendre conjecture and it became known as the Prime Number Theorem (PNT). Nevertheless, this theorem remains approximately true until Riemann Hypothesis is proved for the additive counting error term supplied by complex Riemann zeta function. However, when the argument is a complex number, Riemann zeta function diverges. By analytic continuation, Riemann then defined the values of the complex zeta function in the half-plane x £ 1 and expressed his still unproven hypothesis that all the zeros in the strip 0 £ x £ 1 lie on the line x = ½. For the PNT to be exact, Riemann Hypothesis must be true.