Complex Numbers. Lattice Simulation and Zeta Function Applications

134 Pages·2007·3.772 MB·English

by  S.C. Roy (Auth.)

COMPLEX NUMBERS
Lattice Simulation and
Zeta Function Applications
Stephen C. Roy
Mathematics and Science Tutor
Harrison Allen Education Services
Earlsfield, London, SW18 3DB
Horwood Publishing
Chichester
"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.
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 V-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
VI Preface
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 certain parts of the theory
presented in Chapters 3 and 4. Particularly important regions of the text have been
placed in text boxes - some of these capable of initiating further mathematical or
historical research. Cited references are mostly written in English, with -30%
appearing in other European languages. Several of the more significant documents
are written in German. The bibliography section also includes some extended
descriptions of references that have been identified as central to the development of
important ideas within the text.
Acknowledgements
I would like to thank the following people and organisations for their support
throughout the production of this monograph. Graham Hoare for valuable
correspondence associated with the mathematics of zeta functions and, in
particular, for assistance with derivation of some of the lattice summation
formulae that can be found in Appendix 1. Preparation of Sections 3.3 and 3.4
on contour integration of the xi function was made easier by a useful
communication received from Roger Heath-Brown. Informative material
kindly donated by Walter Purkert, Emilio Elizalde and Samir Khan has also
been helpful. Continued support from Harrison Allen Education Services Ltd,
London, has been very rewarding and I would also like to acknowledge the
European Mathematical Society and the IMA. Finally my thanks are owed to
Francesca Bonner of Horwood Publishing for guidance during final
preparation of the manuscript.
Preface vu
DEPENDENCE CHART
There is some overlap between the sections that are typed in bold and underlined.
Ch .2
2.1
1 1
2.3
2.4
2 8
Ch .3
33 3.2 3.1,3.6
3,5
3A
Ch.4
4.1 ,4.5 4.4
4.3 4.2
Appen dices 1 2
Notations
Some of the following notations aren't defined in the text.
Topic Page numbers
Riemann's zeta function
B Bernoulli numbers
n
B = 1, B, = -0.5, B = 1/6, B = -1/30 28, 42
0 2 4
M (s -1) gamma function (r(s))
r-r Π^>
0
Π ^ Γ Γ ^
s + n
"=1 31,41
Li logarithmic integral
n
f ddtt
Jlog/
22,116
T Re(t) 31
ia, ia Im(t) 22,31
z complex number 41
n natural number 21
x real number 22
Xll Notation
Mangoldt's formulae
θι In or log 44
arctg 44
\72
ij3 arctg 44
η complex parameter 45,46
φ, a complex coefficients 31,40,46
Epstein's zeta function
Z Epstein zeta function 77, 117
g arbitrary lattice vector 77,117
h arbitrary vector 77,117
q(l + g) quadratic form in the components of 1 or 1 + g 77, 117
D a determinant 117
d Res>d 117
s complex variable 77,117
1 lattice vector 77, 117
Ko modified Bessel function 117
k reciprocal lattice vector 117
1 quadratic form defined by an adjoint matrix 117
Lattice simulation
Φ error function
X
2 f 2
φ(χ) = —== I e x dx 94, 111
Π0) reciprocal space lattice potential 88
Π(2) real space lattice potential 79
R,R, real space vectors 79
r radius vector 79
a axis vector 79
f spread vector 80
Ko spread constant 81
q phase velocity 81
ω frequency 81
a ranges from Vq -> co 83
η potential 83
ζ,Ε charge 79,83
h reciprocal space vector
Quantum field theory
S fermionic series 38
F
f(a) a function defined in the complex a plane 38
m natural number 38
s complex number 38
1
Introduction
1.1 COMPLEX NUMBERS
Roots of equations that are neither completely real nor completely imaginary are
often termed complex. By using the word imaginary, reference is made to roots of
negative numbers with V-l being the most popular example.
1.1.1 Early history
Complex numbers have a history that can be traced to work by Greek
mathematician, Heron of Alexandria, who lived sometime between 100 BC and
100 AD. They first appeared in a study concerned with the dimensions of a
pyramidal frustum. Although Heron of Alexandria recognised the conceptual
possibility of negative numbers possessing square roots, it took a considerable
period of time before they started to become of practical significance. This was owed
to discoveries made by Scipione del Ferro and Girolamo Cardano roughly between
1450 and 1600 AD. From 100 AD to the fifteenth century, very little information on
imaginary numbers was recorded. Worthy of note are contributions made by
scholars such as Diophanrus of Alexandria (circa 300 AD) and Mahaviracarya (circa
850 AD) who both also considered the conceptual possibility of square roots of
negative numbers. By the eighteenth century complex numbers had achieved
considerable recognition and were starting to become written as, for example, 3 + 5i,
where 3 represents what is known as the real component and 5i is the imaginary
component. The letter / is representative of V-l and was first used by Euler in 1777.
To this day they have been expressed in this manner.
A considerable portion of this book relies on a geometric consideration of complex
numbers as opposed to an algebraic one. The geometry can be understood by
consideration of work by Wallis, Wessel and Argand which spans from the
seventeenth to the nineteenth century. This work was responsible for what are now
2 Introduction [Ch. 1
commonly known as Argand diagrams that represent complex numbers by an
imaginary y-axis and a real x-axis as illustrated in Fig. 2.2.1 for example. The first
notion that the y-axis should be positioned vertically with respect to the real axis was
provided by Wallis (1616-1703). For further information on the history of complex
numbers Nahin (1998) provides a clear account in a recent publication.
1.1.2 Complex function theory
Important ideas associated with this aspect of complex number theory were first
conceived by Augustin-Louis Cauchy (1789-1857) in a paper written in 1814 that
describes the integration of complex functions. Gauss was involved with similar
work at the same time that Cauchy published these findings. Riemann made use of
Cauchy's work in his doctoral dissertation in 1851. In Sections 3.3 and 3.4 complex
function integration is studied in some detail with reference to the xi function
defined by Riemann.
1.1.3 Practical applications
Complex numbers find widespread use in many scientific subject areas. They can be
applied in many branches of physics and also in astronomy to monitor planetary
motion. Electrical engineers use complex numbers to assess and evaluate electronic
circuitry and any science student is likely to first encounter applications of complex
numbers in this context as well. Within electrochemistry, which is a branch of
chemistry often concerned with characterisation and optimisation of electrical
devices such as batteries, fuel cells and sensors, the complex algebra linked to AC
impedance theory can be used to rationalise ways in which devices perform as they
operate (Roy, 1996). For example, the manner in which a battery generates power
can be modelled by a combination of resistors and capacitors. The AC impedance of
this model replicates the battery performance and therefore provides a deeper
understanding of the associated chemical mechanisms that take place.
1.2 SCOPE OF THE TEXT
In Chapter 2, an outline of theory used throughout the monograph is provided. The
chapters that follow from this are presented primarily to provide descriptions of
previous mathematical and scientific investigations that have involved the use of
complex numbers. As each of these chapters evolved, new ideas inclined towards
research were conceived and duly incorporated at appropriate parts of the text. Three
research papers are central to Chapters 3 and 4. These are:
B. Riemann: Monatsberichte der Berliner Akademie, 1859, p671
H. von Mangoldt: Mathematische Annalen, 60, 1905, pi
P.P. Ewald: Annalen der Physik, 64, 1921, p253
Chapter 3 consists of a study of the application of complex analysis in number
theory with respect to Riemann's zêta function and is composed of three main areas.
The first of these is mainly introductory with some discussion of the functional
equation of the zeta function and line integration processes that are used to derive it.
Line integration is one of the first mathematical tricks that must be understood