Analytic Number Theory for Undergraduates

132 Pages·2009·35.87 MB·English

by  Heng Huat Chang

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Monographs in Number Theory ISSN 1793-8341
Series Editors: Bruce C. Berndt
(University of Illinois at Urbana-Champaign, USA)
Heng Huat Chan
(National University of Singapore, Singapore)
Editorial Board Members:
Jonathan M. Borwein (The University of Newcastle, Australia)
William Duke (University of California at Los Angeles, USA)
Wen-Ching Winnie Li (The Pennsylvania State University, USA)
Kannan Soundararajan (Stanford University, USA)
Wadim Zudilin (The University of Newcastle, Australia)
Published
Vol. 1 Analytic Number Theory — An Introductory Course
by Paul T. Bateman & Harold G. Diamond
Vol. 2 Topics in Number Theory
by Minking Eie
Vol. 3 Analytic Number Theory for Undergraduates
by Heng Huat Chan
LaiFun - Analytic Num Theory.pmd 2 3/11/2009, 2:15 PM
Heng Huat Chan
National University of Singapore, Singapore
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
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Acquisition Editor: Dr Phua Kok Khoo
Desk Editor: Kwong Lai Fun
In-house Series Editor: Tan Siew Lian
Designer: Mayrine Chung Mei Lan
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ANALYTIC NUMBER THEORY FOR UNDERGRADUATES
Monographs in Number Theory — Vol. 3
Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-981-4271-35-6
ISBN-10 981-4271-35-7
ISBN-13 978-981-4271-36-3 (pbk)
ISBN-10 981-4271-36-5 (pbk)
Printed in Singapore.
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February13,2009 16:7 WorldScientificBook-9inx6in AnalyticalNumberTheory
This book is dedicated to my professors, classmates and friends
at the University of Illinois at Urbana-Champaign
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Preface
I took several mathematics courses as a graduate student at University of
Illinois at Urbana Champaign and one of the most interesting courses is
A. Hildebrand’s course titled “Analytic Number Theory 1”. With Hilde-
brand’s lecture notes and T.M. Apostol’s book [2] as references,I designed
an undergraduate level course on “Analytic Number Theory” and taught
the course for several years between 2001 and 2007 at the National Uni-
versity of Singapore. The currentbook is a collectionof notes arisingfrom
thiscourse. Thebookissuitableforadvancedundergraduatestudentswho
wish to understand number theory beyond elementary level.
There are seven chapters in this book. The materials in Chapter 1 are
forreaderswhoarenotfamiliarwithelementarynumbertheory. Themain
focus of Chapter 2 is arithmetical functions while Chapter 3 discusses the
averagesofarithmeticalfunctions. Elementaryresultsrelatedtothe Prime
Number Theorem are studied in Chapter 4 and the proof of the Prime
Number Theorem is presented in Chapter 5.
Chapter 6 is devoted to the study of Dirichlet’s series and the proof of
Landau’sTheorem. Thesematerialsareessentialintheconcludingchapter,
wherebyDirichlet’sTheoremonprimesinarithmeticprogressionisproved.
I would like to thank A. Hildebrand for his permission to include part
ofhis lecture notes inthis book. I amalsoverygratefulto B.C.Berndtfor
his advice on the preliminary version of this book.
I started writing this book in 2007. The final version was completed in
Nagoya University when I was a Hitachi Fellow. I would like to thank the
Hitachi Scholarship foundation for their generous support and my host at
Nagoya University, Y. Tanigawa,for his warm hospitality.
Heng Huat Chan
24 December 2008
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