PNT is the acronym for the Prime Number Theorem. But before giving a precise statement, its relation to the base of natural logarithm must be described. On the one hand, the base of the natural logarithm defines an area within the Cartesian coordinate system bounded by the equilateral hyperbola xy=1, the x-axis, and the two vertical lines x=1 and x=N. This area is simply logN to the natural base of logarithm. As early as 1792 Gauss observed that the distinct counts of prime numbers among the integers up to the upper bound integer N denoted by p(N) is approximately equal to the ratio of N over logN: p(N)=N/logN. At the time of his discovery a proof was not provided. It was simply his conjecture. Later, Legendre was able to improve its proper formulation. Nonetheless, the proof was finally provided simultaneously by both Hadamard and de la Vallée Poussin in 1896 applying the theory of entire functions of complex variables.

It can now be realized that the necessity of using the analytic functions of complex variables is because of the implied fractal geometry and dimension of N/logN. If the highest dimensionality of N is one and that of logN is two then the highest dimensionality of N/logN is 1/2 hence the random chaotic pattern of prime numbers. This is truly unnecessary since within the sieve of Diophantus the topology of PNT is absolutely an ordered matrix structure. It has a fixed dimensionality of two and the patterned locations of the prime numbers are completely and totally predictable.