Complex Numbers: Lattice Simulation and Zeta Function Applications

by S C Roy

Overview

An informative and useful account of complex numbers that includes historical anecdotes, ideas for further research, outlines of theory and a detailed analysis of the ever-elusory Riemann hypothesis. Stephen Roy assumes no detailed mathematical knowledge on the part of the reader and provides a fascinating description of the use of this fundamental idea within the two subject areas of lattice simulation and number theory. Complex Numbers offers a fresh and critical approach to research-based implementation of the mathematical concept of imaginary numbers. Detailed coverage includes:

• Riemann’s zeta function: an investigation of the non-trivial roots by Euler-Maclaurin summation.
• Basic theory: logarithms, indices, arithmetic and integration procedures are described.
• Lattice simulation: the role of complex numbers in Paul Ewald’s important work of the I 920s is analysed.
• Mangoldt’s study of the xi function: close attention is given to the derivation of N(T) formulae by contour integration.
• Analytical calculations: used extensively to illustrate important theoretical aspects.
• Glossary: over 80 terms included in the text are defined.

• Offers a fresh and critical approach to the research-based implication of complex numbers
• Includes historical anecdotes, ideas for further research, outlines of theory and a detailed analysis of the Riemann hypothesis
• Bridges any gaps that might exist between the two worlds of lattice sums and number theory

Product Details

ISBN-13:	9780857099426
Publisher:	Elsevier Science
Publication date:	07/01/2007
Sold by:	Barnes & Noble
Format:	eBook
Pages:	148
File size:	4 MB

About the Author

Dr. Stephen Campbell Roy from the green and pleasant Scottish town of Maybole in Ayreshire, received his secondary education at the Carrick Academy, and then studied chemistry at Heriot-Watt University, Edinburgh where he was awarded a BSc (Hons.) in 1991. Moving to St Andrews University, Fife he studied electro-chemistry and in 1994 was awarded his PhD. He then moved to Newcastle University for work in postdoctoral research until 1997. Then to Manchester University as a temporary Lecturer in Chemistry to teach electrochemistry and computer modelling to undergraduates.

Table of Contents

• Dedication
• About our Author
• Author’s Preface
  • Background
  • Important features
  • Acknowledgements
  • DEPENDENCE CHART
• Notations
• 1. Introduction
  • 1.1 COMPLEX NUMBERS
  • 1.2 SCOPE OF THE TEXT
  • 1.3 G. F. B. RIEMANN AND THE ZETA FUNCTION
  • 1.4 STUDIES OF THE XI FUNCTION BY H. VON MANGOLDT
  • 1.5 RECENT WORK ON THE ZETA FUNCTION
  • 1.6 P. P. EWALD AND LATTICE SUMMATION
• 2. Theory
  • 2.1 COMPLEX NUMBER ARITHMETIC
  • 2.2 ARGAND DIAGRAMS
  • 2.3 EULER IDENTITIES
  • 2.4 POWERS AND LOGARITHMS
  • 2.5 THE HYPERBOLIC FUNCTION
  • 2.6 INTEGRATION PROCEDURES USED IN CHAPTERS 3&4
  • 2.7 STANDARD INTEGRATION WITH COMPLEX NUMBERS
  • 2.8 LINE AND CONTOUR INTEGRATION
• 3. The Riemann Zeta Function
  • 3.1 INTRODUCTION
  • 3.2 THE FUNCTIONAL EQUATION
  • 3.3 CONTOUR INTEGRATION PROCEDURES LEADING TO N(T)
  • 3.4 A NEW STRATEGY FOR THE EVALUATION OF N(T) BASED ON VON MANGOLDT’S METHOD
  • 3.5 COMPUTATIONAL EXAMINATION OF ?(s)
  • 3.6 CONCLUSION AND FURTHER WORK
• 4. Ewald Lattice Summation
  • 4.1 COMPUTER SIMULATION OF IONIC SOLIDS
  • 4.2 CONVERGENCE OF LATTICE WAVES WITH ATOMIC POSITION
  • 4.3 VECTOR POTENTIAL CONVERGENCE WITH ATOMIC POSITION
  • 4.4 DISCUSSION AND FINAL ANALYSIS OF THE EWALD METHOD
  • 4.5 CONCLUSION AND FURTHER WORK
  • APPENDIX 1
  • APPENDIX 2
• Bibliography
• Glossary
• Index