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:
Published by Chiron Academic Press – Sweden
ISBN 978-91-7637-197-8
Chiron Academic Press is a Wisehouse Imprint.
© Wisehouse 2016 – Sweden
© 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
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
