The Secret Harmony of Primes. Sam Vaseghi

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      The Secret Harmony

       of Primes

      The Secret Harmony

       of Primes

      by

      Sam Vaseghi

      Chiron Academic Press

       Sam Vaseghi

      The Secret Harmony of Primes

      You are welcome to contact the author’s desk via:

       [email protected]

      Published by Chiron Academic Press – Sweden

      ISBN 978-91-7637-197-8

      Chiron Academic Press is a Wisehouse Imprint.

      © Wisehouse 2016 – Sweden

       www.wisehouse-publishing.com

      © Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photographing, recording or otherwise), without the prior written permission of the publisher.

      For Darya and Soheila

      “The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.”

      Henri Poincaré, Science and Method

       1865-1936

      Contents

       The fundamental theorem of arithmetic

       The log-canonical form

       CHAPTER 2

       Structures of Harmony

       The prime exponent theorem

       The prime power modulo functions

       (a) The relation to the Euler Gamma function

       (b) The trivial relation to the function

       The elementary prime exponent functions

       CHAPTER 3

       Secrets of Design

       The curve family over the prime exponent vectors

       Scaling in a metric space

       CHAPTER 4

       The Spectrum

       The reconstruction of the log-canonical form

       Certainty versus uncertainty

       The orbits

       The spectrum of natural numbers

       The log-canonical form for

       CHAPTER 5

       Zeta

       CHAPTER 6

       The Recursive Sequence of the Primes

       The implicit recursive sequence of the primes

       On the Number of Primes Less Than a Given Magnitude

       The implicit recursive sequence of the primes

       The interference prime counting function

       How fair are the prime number dice of Gauss?

       NOTES

      THE SEQUENCE OF THE PRIMES IS SOMETHING OF A MYSTERY in number theory, if not in all of mathematics. Hardly any topic has ever fascinated mathematicians more.

      Leonhard Euler once said, “Mathematicians have tried in vain, to this day, to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate”.1

      One key question has always been: Does there exist a single, overarching organisational principle that accounts for why any prime follows another prime as the next largest prime in the prime sequence?

      The American mathematician Don Bernard Zagier wrote, “Despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict

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