Page xi - Prologue

      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

Page x - Prologue

      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

Page ix - Prologue

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

Page viii - Contents

4. A New Kind of Action - Page 347

5. Complex Conformal Mappings - Page 349

6. Complex Angles - Page 353

7. Gudermannian Action - Page 359

18. Riemann Hypothesis – The Classical Form - Page 365

1. The Unfolding of the Abelian Functions - Page 367

2. Riemann Hypothesis – The Classical Form - Page 384

Epilogue - Page 391

Appendix - Page 399

1. Distribution of Primes – Fourth Interval [1, 2310] - Page 401

2. Distribution of Primes – Fifth Interval [1, 30300] - Page 406

       3. First Six Gap Sequences - Page 456

Page vii - Contents

11.     Circular Functions versus Hyperbolic Functions - Page 243

1.    The Circle and the Hyperbola - Page 245

2.    Circular Angles – Circular Radians - Page 246

3.    Hyperbolic Angles – Hyperbolic Radians - Page 251

12. Hyperbola and some of its Properties - Page 259

1. The Hyperbola in the x-y Coordinates - Page 261

2. Rotation of a coordinate system - Page 262

3. The Hyperbola in the u-v Coordinates - Page 263

4. Squaring a Hyperbola – Part 1 - Page 264

5. Squaring a Hyperbola – Part 2 - Page 266

6. Hyperbola and the Arithmetic/Geometric Progressions - Page 268

7. Squaring a Hyperbola – Part 3 - Page 270

13. The Gudermannian Angles - Page 273

1. The Gudermannian Angles - Page 275

2. Geometrical Interpretation of the Gudermannian - Page 280

14. Prime Number Theorem - Page 285

1. Some Trigonometry - Page 287

2. The Prime Number Theorem – A Proof - Page 291

15. Riemann Hypothesis – Equivalent For - Page 299

1. Riemann Hypothesis – A Proof - Page 301

16. Trigonometric Generalizations - Page 327

1. A Generalization of De Moivre’s Theorem - Page 329

2. Multiple Angles Relationships - Page 333

17. Harmonic Actions - Page 335

1. Circular Action - Page 337

2. Hyperbolic Action - Page 339

3. Exponential Action - Page 341

Page vi - Contents

3.    Method for Generating Gap Functions of Second Degree - Page 105

4.    Gap Functions of Third Degree - Page 107

5    Gap Functions of Forth Degree - Page 112

6    Riemann Theta Functions for Natural Numbers - Page 113

6     Gap Sequences Generation in the Complex Domain - Page 119

1.    Roots of Unity - Page 121

2.    Simple Periodic Complex Functions - Page 122

3.    Generating Primes in the Complex Domain - Page 124

7     A Topological View - Page 133

1.    A Very Interesting Property of the Hypocycloids - Page 135

2.    Doubling the Cube - Page 137

3.    Eratosthenes’ Construction - Page 140

4.    From Archytas Construction to Abelian Functions - Page 142

5.    A Topological View of the Conical Sections - Page 156

8     Prime Numbers and Topology - Page 159

1.    Eratosthenes’s Sieve on a Riemann’s Surface of Genus g - Page 161

2.    Closed Lines on the Archytas’ Torus - Page 173

3.    The Complex Logarithm - Page 182

9     Möbius Functions - Page 185

1.    Möbius Function vs Liouville Function - Page 187

2.    Möbius Functions - Page 191

3.    Möbius Patterns - Page 205

4.    Probability that a Number is Square Free - Page 215

10   Riemann’s Theta Function for s=1/2 - Page 217

1.    An Interesting Function - Page 219

2.    Pascal’s Triangle - Page 227

3.    Leibnitz Harmonic Triangle - Page 233

4.    Riemann’s Theta Function for s = ½ - Page 237

5.    The Meaning of ζ(1/2) - Page 240

Page v - Contents

Contents

Prologue - Page ix

1.     Primes Factorial Intervals - Page 001

1.    Eratosthenes’s Sieve - A Geometrical Approach - Page 003

2.    Periodic Behaviour – First Observation - Page 005

3.    Symmetry – Second Observation - Page 008

2.     Gaps Sequences - Generating Primes - Page 015

1.    Gap Functions of First Degree - Page 017

2.    Properties of Gap Functions of First Degree - Page 022

3.    Fractal like Behaviour of Gap Functions - Page 025

4.    Recursive Method for Generating Gap Functions – Case Study - Page 026

5.    Recursive Method for Generating Prime Numbers - Page 028

6.    A New Life to Euclid’s Prime Test - Page 040

7.    The numbers of Primes between n and n² - Page 046             

8.    The number of Primes between n and 2n - Page 048

3.     More than Statistics - Page 053

1.    Histograms of Gap Functions - Page 055

2.    Couple Functions - Page 057

3.    Gap Pairs Sum Functions - Page 062

4.    Gap Functions and Riemann’s Zeta Function for s = 1 - Page 067

5.    Gap Functions and Step Functions - Page 069

6.    Other Results from Gap Sequences - Page 076

7.    Patterns in the Distribution of the Number of Factors - Page 077

8.    The Histogram of Gap Functions and the Blackbody Radiation - Page 083

4.     Big Gaps between Primes - Page 087

1.    Chinese Reminder Theorem – Where are the big gaps? - Page 089

5.     Gap Functions of Higher Degrees - Page 097

1.    Gap Functions of Second Degree - Page 099

2.    Properties of Gap Functions of Second Degree - Page 102

Page iv - Blank Page

This page is intentionally left blank.

Page iii - Dedication

To my wife Cristina,
and our children,
Jason and Nicole.

"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"
David Hilbert

“Yes!”
Nick Trif

Page ii - Copyright

Copyright 2010 by Nick Trif

Ottawa, Canada
All rights reserved.

This work has been registered with the Canadian Property Office
Registration Number 1079862

ISBN 0-9866046-1

Page i - Front Cover

Complex Arithmetic

Patterns Behind Numbers

The inner working of an evolving Universe

by Nick Trif