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      universe is to use transcendental function. As Gauss, Abel, Jacobi and Riemann have shown, there is an extended class of transcendental functions, best known today as elliptical functions and abelian functions. The use of the transcendental functions has to be combined with the method of inversion. When Jacobi was asked about how he has made so many discoveries, he said: “Always invert.” Using this method he made most of the work of Lagrange on elliptical integrals obsolete.

5.      Try to understand how the most remarkable results known, relate to your problem and how were them discovered. Try to understand the context and the state of mind of the persons that made those discoveries. It seems obvious to me that knowing as much as possible about Riemann will only help in trying to prove the Riemann Hypothesis. If we can get to the point to re-make or re-discover the hypothesis, probably we are very close to where Riemann was when he stated it in the first place. Also, use the original work to get the original ideas. Copies of an idea most of the time become distorted; they are not conformal maps of the original idea. Go back to the titans and get their ideas and their methods. These are proven to work and will probably take you to the solution.

6.      Start working, be consistent, patient and enjoy what you are doing. Don’t wait until you master the subject. Again, it was Jacobi who said: “Your father would never have married and you wouldn’t be here now, if he had insisted in knowing all the girls in the world before marrying one.”

The order chosen to present the ideas and the finding in this book is a combination of both, chronologic and logic. The initial chapters are mostly presented in a chronological order – this is the way I investigated the problem. The results of the investigation are presented in a logical order: at least in that logic that makes sense in my mind.

“I will not make poems with reference to parts;

But I will make leaves, poems, poemets, songs, says, thoughts, with

reference to ensemble:

And I will not sing with reference to

a day, but with reference to all

days;”

Walt Whitman

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      continuum manifold. The relationships of this manifold obey the same general laws as any multiple extended manifolds with the same number of magnitudes. The specific manifestations of these relationships can only be determined by experience. Any problem is basically a “geometric problem”, or better said a “hyper-geometric problem” in its specific multiple extended manifold. This was the approach taken by Archytas, Leibnitz, Gauss and Riemann, to name just the most remarkable men of science that come to my mind now.

2.      Start from a good set of experimental data. I learned this from Kepler. It was his luck (and ours) that he got all the astronomical recordings of Tache Branch to work with, recordings taken in uncounted nights over many years. As a general rule start from this data and go back to it to verify any assumption you have made and also to look for new ideas about futher investigations and new directions of attack. Don’t forget that it is in this data that the pattern you are looking for is hidden. For us, the humans, the numbers are always available to us. Our mind is able to count, or inversing “the counting is a characteristic of the human mind”. Again, I will argue, any pattern that could possible exist can also be found as a pattern of numbers. From this perspective it is so easy to understand Pythagoras’ and his insight that everything that exists is a number. It is our advantage that today, using computers in the right way, we have the tools to test and verify our assumptions faster than in any time in human history. But it is also true that it is so easy to be lost in details, to lose focus, to concentrate on the wrong approach and most of the time to not be able to see the big picture. This requires a good understanding of the computer as a tool. It shall never be forgotten that the study of natural phenomena is the main purpose of the investigation.

3.      Use the complex domain. Don’t forget that in the last 150 years all new true discoveries came when using the complex domain in solving problems. Again, this can be explained by at least two reasons. First, all the simple patterns, the one that can be seen and understood using simple mathematics have already been discovered. There are tens of centuries of recorded history. There is no reason to believe that people who lived thousand or many hundreds of years ago were not as “smart”, as “evolved”, as “curious” or as dedicated to their work as we are. This is why all the simple problems and patterns have already been discovered. Second, the complex domain has proved to be the place from where all problems can be seen in a totally new light and from where apparently strange and unexplained features can be easily explained. The complex domain, its closure and its geometrical representation using surfaces, opens the door to the true and complete understanding of the Universe. That is why I believe a better name for the “complex domain” would be the “complete domain”.

4. Use transcendental functions and inversion. It was Leibnitz who said that the discovery of the exponential function (logarithm), the main actor of the first class of transcendental functions, has ended the use of algebraic functions in the study of natural phenomena. The only way to explain the rich manifestation of the physical

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Prologue

“If you have an apple and I have an apple and we exchange these apples then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas.”

George Bernard Shaw

My fascination with prime numbers started before I can remember. I was always sure there must be patterns governing their distribution and also I felt that the understanding of these patterns was in my grasp.

I knew I was in good company. Over the millennia all the great mathematicians have tried to find these patterns, and most of them felt that there must be such patterns. It was Don Zaghier in his well known paper "The first 50 Million Prime Numbers" that said it best:

"There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts.

The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law then that of chance, and nobody can predict where the next one will sprout.

The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with the almost military precession."

By taking on such a well known problem I proposed to myself to find first the best approaches that could lead me to a solution. I was kind of aware of the difficulty and complexity of the problem, but the real implication and the depth of the solution only revealed to me after long and continuing meditation and profound immersion of all my thoughts into the problem during many years.

Looking back in history to men with the most relevant contribution to the development of mathematics and science, few methods and principles, applied by them, stand out. Following them, these are the main guidance principles I followed during my investigations.

1. Take a geometrical approach. It is always better to use a geometrical construction when trying to explain a complicated phenomena or trying to understand a complicated pattern. There are at least two reasons that make this a better approach. First, we experience life in a multidimensional space that exposes us to a richer variety of patterns and phenomena than those available in a linear type, formal approach. Our brains have developed to cope with the complexity of the multidimensional space. I will argue that there is no better method to communicate complex ideas then geometry. Second, the space as shown by Riemann is just one example of a multiple extended