Complex Numbers: Lattice Simulation and Zeta Function Applications

By S C Roy

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

• Language: English
• Publisher: Elsevier Science
• Release date: Jul 1, 2007
• ISBN: 9780857099426

S C Roy

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.

Book preview

Complex Numbers

Lattice Simulation and Zeta Function Applications

First Edition

Stephen C. Roy

Mathematics and Science Tutor, Harrison Allen Education Services, Earlsfield, London, SW18 3DB

Horwood Publishing

Chichester

Table of Contents

Cover image

Title page

Dedication

About our Author

Copyright page

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 ζ(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

Dedication

"Mathematics possesses not only truth, but supreme beauty – a beauty cold and austere like that of sculpture, and capable of stern perfection, such as only great art can show.

Bertrand Russell (1872-1970) The Principles of Mathematics

About our Author

Dr. Stephen Campbell Roy from the Scottish town of Maybole in Ayrshire, received his secondary education at 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 Research Associate in materials science.

Stephen’s postdoctoral experience includes:

a) Research-related lecturing at Newcastle, Manchester, Bath and London Universities.

b) Mathematics teaching to A-Level standard.

c) Team leader at Daresbury Laboratory in Warrington, Cheshire for synchrotron radiation research.

d) Prize-winner for his presentation on chemistry research in London, 1995.

e) Royal Society grant towards attendance at the 10th International Conference on Solid State Ionics at Singapore, 1995.

Stephen is currently science and mathematics tutor for Harrison Allen Education Services in London. He became aware of the interactive link between mathematics and chemistry after realising the importance of mathematics for carrying out chemical experiments. His membership of the Institute of Mathematics and its Applications and of the European Mathematical Society as Council Delegate has helped to broaden his mathematical horizons in number theory. A former Heriot-Watt colleague commended Stephen as a potential author for the Horwood Publishing Series Mathematics and Its Applications, which has led to the publication of this book.

Copyright

HORWOOD PUBLISHING LIMITED

International Publishers in Science and Technology

Coll House, Westergate, Chichester, West Sussex,

PO20 3QL England

www.horwoodpublishing.net

First published in 2007

COPYRIGHT NOTICE

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the permission of Horwood Publishing, Coll House, Westergate, Chichester, West Sussex, PO20 3QL England

© S C Roy, 2006

ISBN-10: 1-904275-25-7

ISBN-13: 978-1-904275-25-1

British Library Cataloguing in Publication Data

A catalogue record of this book is available from the British Library

Printed and bound in England by Antony Rowe Ltd, Chippenham

Author’s Preface

Background

The concept of the complex number was first presented to me during a short course on AC electricity that formed part of a physics syllabus. Instruction provided on the use of imaginary numbers, based on √-l, helped to explain a multitude of electrical characteristics associated with inductors, capacitors and resistors. Further studies of complex algebra that formed part of a first year undergraduate mathematics course and the use of Euler’s identities in a supplementary physics course provided a different perspective with the introduction of the exponential function. Later participation in a project aimed at the optimisation of electrochemical cells suitable for the production of alkali metal beams for semiconductor manufacture demonstrated again that complex analysis was a topic that required some consideration. This was owed to extensive use of electrical instrumentation designed to measure, amongst other things, resistances of materials used for manufacture of the cells.

Involvement with research in the computer modelling of ionic solids also introduced Ewald’s lattice summation method. A detailed study of the original paper again indicated the importance of complex number theory and is largely responsible for the production of this monograph. Chapter 4 consists of a scientific approach to the discussion of Ewald’s method that relies on the use of graphs to identify and illustrate important theoretical aspects. Related mathematical literature has also provided an introduction to analytic number theory and zeta functions. The study of energies within crystals has a history that is inextricably linked to mathematical investigations associated with these functions and a large amount of complex number work is tied to this branch of mathematics as well. Throughout this monograph, Riemann’s zeta function is described in detail and a considerable amount of information on ideas that are central to Riemann’s hypothesis (RH) is presented with contour integration processes described.

Important features

While writing this account the following three objectives were considered. Firstly the production of an adequate translation of Ewald’s method; second, clear outlines of theory associated with the zeta function; thirdly an emphasis on possible methods of analysing or even solving Riemann’s hypothesis. Although primarily a descriptive textbook, short sections with suggestions for further work have been included to point the way for research-based continuations. In several instances it can be regarded as a book of contradictions with questions raised during studies of previous work by Ewald, von Mangoldt and also Edwards. The dependence chart that I have included in this preface provides some indication of the inter-activity of the various sections. Most notably the linkage of relevant theory to the foci of Chapters 3 and 4. There is also some overlap between Sections 3.3, 3.4 and 3.5, with reference to equations and figures that are not always confined to the section of interest. Throughout the chapters certain terms are either highlighted in bold or written in italics. The former mainly represent important new concepts defined in the glossary while the latter correspond to ideas that certain students should already be familiar with.

The style is informal, with an absence of open questions and revision exercises. However close attention should be given to the questions and answers that are located in Chapter 2 either before or during an analysis of