G.H. Hardy (1877-1947), see also http://en.wikipedia.org/wiki/G._H._Hardy, was one of the great number theorists of the early 20th century. He was instrumental for introducing the great mathematician Ramanujan into the mainstream of modern mathematics. Hardy was also an eloquent promoter and made 2 notable comments:

Verbatim, the latter is as follow: “The theory of numbers, more than any other branches of pure mathematics, has begun by being empirical science. Its most famous theorems have all been conjectured, sometimes a 100 years or more before they have been proved, and they have been suggested by the evidence of a mass of computation,” Hardy 1920.

The second 5 years earlier comment not verbatim is that “In the analytical investigation of prime numbers theory anyone can make plausible conjectures, and they are almost always false,” Hardy 1915.

The 2nd comment holds the key to a complete understanding of prime numbers. The question is why should any number theory be analytical? It is more questionable since analysis implies breaking things into jigsaw pieces. Impossible solutions arise where and when pieces from different sets are intentionally mixed together, for example, mixing real and imaginary numbers. On the contrary, synthesis implies piecing them into a singular whole, for example, rearranging a mixture of even, odd, and prime numbers.

The fundamental problem with the Riemann Hypothesis of complex zeta function is that it mixes real and imaginary numbers to find an additive factor to Gauss Prime Number Theorem. A return to Euler real zeta function as the union of two disjoint sets: [0, 1) and (1, ∞) for z(s) where s=2m+3n is not necessary by introducing an open Diophantine sieve which is simply the set of whole numbers minus unity: 0, 2, 3, 4, 5, 6, 7, 8, …, ∞ while the sieve indices m and n take on values 0, 1, 2, 3, 4, 5, 6, 7, 8, …, ∞. For n=1, and m varies from 0 to ∞ all the prime numbers can be found in a singular vertical column of the open Diophantine sieve. Goldbach conjecture can be proved true merely by using the sieve indices and a simple algorithm for finding prime factors of large number is also hidden within the open Diophantine sieve.