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