Analytic Number Theory for Undergraduates PDF
Preview Analytic Number Theory for Undergraduates
7252tp.indd 2 3/13/09 11:44:32 AM 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 7252tp.indd 1 3/13/09 11:44:29 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. 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. LaiFun - Analytic Num Theory.pmd 1 3/11/2009, 2:15 PM 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 v February13,2009 16:7 WorldScientificBook-9inx6in AnalyticalNumberTheory 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 vii
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