Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics) (Hardcover)

~ Tom M. Apostol (Author) "The theory of numbers is that branch of mathematics which deals with properties of the whole numbers, 1, 2, 3, 4, 5, . . ..." (more)
Key Phrases: induced modulus, induced moduli, bounded partial sums, Dirichlet L-functions, Shapiro's Tauberian, Definition Let (more...)

Editorial Reviews

Review

T.M. Apostol Introduction to Analytic Number Theory "This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages. The presentation is invariably lucid and the book is a real pleasure to read." —MATHEMATICAL REVIEWS

Product Description

This introductory textbook is designed to teach undergraduates the basic ideas and techniques of number theory, with special consideration to the principles of analytic number theory. The first five chapters treat elementary concepts such as divisibility, congruence and arithmetical functions. The topics in the next chapters include Dirichlet's theorem on primes in progressions, Gauss sums, quadratic residues, Dirichlet series, and Euler products with applications to the Riemann zeta function and Dirichlet L-functions. Also included is an introduction to partitions. Among the strong points of the book are its clarity of exposition and a collection of exercises at the end of each chapter. The first ten chapters, with the exception of one section, are accessible to anyone with knowledge of elementary calculus; the last four chapters require some knowledge of complex function theory including complex integration and residue calculus.

Product Details

• Hardcover: 352 pages
• Publisher: Springer (May 28, 1998)
• Language: English
• ISBN-10: 0387901639
• ISBN-13: 978-0387901633
• Product Dimensions: 9.4 x 6.5 x 0.8 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (6 customer reviews)
• Amazon.com Sales Rank: #217,618 in Books (See Bestsellers in Books)

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34 of 36 people found the following review helpful:

5.0 out of 5 stars Unsurpassed SECOND text on number theory, June 28, 2004

By	rjohnp "rjohnp" (Beaverton, Oregon United States) - See all my reviews

The amazing positives of this book are accurately described in the other reviews so I will skip them. There is no negative, but the other reviewers assert that the reader needs no prior exposure to number theory. I completely disagree.

While this book does quickly cover elementary number theory, a reader new to this field will quickly feel lost. Without more exposure and a good prior feel for elementary number theory, the use of analytic techniques will seem ad hoc instead of following a logical pattern. By way of example, three areas covered in this book that are not part of analytic number theory and for which the reader would do better to learn from a less sophisticated text are the Fermat-Euler Theorem, Diophantine equations, and quadratic reciprocity.

Excellent texts for a first exposure to number theory are, from simpler to more difficult:

1. Elementary Number Theory by Underwood Dudley

2. An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery

3. An Introduction to the Theory of Numbers by Hardy and Wright

Apostol's book on analytic number theory is a classic that may never be surpassed. It is a marvelous second book on number theory.

20 of 20 people found the following review helpful:

5.0 out of 5 stars Excellent exercises in a clear exposition, November 12, 2000

By A Customer

This book has excellent exercises at the end of each chapter. The exercises are interesting and challenging and supplement the main text by showing additional consequences and alternate approaches.

The book covers a mixture of elementary and analytic number theory, and assumes no prior knowledge of number theory. Analytic ideas are introduced early, wherever they are appropriate. The exposition is very clear and complete. Some novel features include: three chapters on arithmetic functions and their averages (including a simple Tauberian theorem due to Shapiro); Polya's inequality for character sums; and an evaluation of Gaussian sums (by contour integration), used in one proof of quadratic reciprocity.

36 of 40 people found the following review helpful:

5.0 out of 5 stars well presented, delightfully written, December 5, 2001

By	Dr Brown (Hong Kong) - See all my reviews

I think that there will be little harm if the title of the book is changed to 'Introduction to elementary number theory' instead. The author presumes that the reader has not any knowledge of number theory. As a result, materials like congruence equation, primitive roots, and quadratic reciprocity are included.
Of course as the title indicates, the book focusses more on the analytic aspect. The first 2 chapters are on arithmetic functions, asymptotic formulas for averaging sums, using elementary methods like Euler-Maclaurin formula .This lay down the foundation for further discussion in later chapters, where complex analysis is involved in the investigation. Then the author explain congruence in chapter 4 and 5. Chapter 6 introduce the important concept of character. Since the purpose of this chapter is to prepare for the proof of Dirichlet's theorem and introduction of Gauss sums, the character theory is developed just to the point which is all that's needed. ( i.e. the orthogonal relation). Chapter 7 culminates on the elementary proof on Dirichlet's theorem on primes in arithmetic progression. The proof still uses L-function of course, but the estimates, like the non-vanishing of L(1) , are completely elementary and is based only on the first 2 chapters.
The author then introduce primitve roots to further the theory of Dirichlet characters. Gauss sums can then be introduced. 2 proofs of quadratic reciprocity using Gauss sums are offered. The complete analytic proof, using contour integration to evaluate explicitly the quadratic Gauss sums, is a marvellous illustration of how truth about integers can be obtained by crossing into the complex domains.
The book then turns in to the analyic aspect. General Dirichlet series, followed by the Riemann zeta function, L function ,are introduced. It's shown that the L- functions have meremorphic continuation to the whole complex plane by establishing the functional equation L(s)= elementary factor * L(1-s). The reader should be familiar with residue calculus to read this part.
Chapter 13 may be a high point of this book, where the Prime Number Theorem is proved. Arguably, it's the Prime Number Theorem which stimulate much of the theory of complex analysis and analyic number theory. As Riemann first pointed out, the Prime Number Theorem can be proved by expressing the prime counting function as a contour integral of the Riemann zeta function, then estimate the various contours. The proof given in this book , although not exactly that envisaged by Riemann , is a variant that run quite smoothly. As is well known , a key point is that one can move the contour to the line Re(s)=1, and to do this one have to verify that zeta(s) does not vanish on
Re(s)=1.The proof , due to de la vale-Poussin, is a clever application of a trigonometric identity. Unfortunately, the method does not allow one penetrate into the region 0<Re(s)<1, where the distribution of zeroes in this region contain the information about the flunctuation of Pi(x) areounf x/log x. The famous Riemann Hypothesis states that the only zeroes in this region lis on the line Re(s)=1/2. After more than 100 years, although the Riemann Hpothesis has natural generalisation to number fields, neither of these RH is proven, which indicates the difficulties of this problem. Recently some new directions, related to quantum statistical mechanics, has been connected with this old problem. If the RH is proven, then the set of prime numbers , although looks completely random locally ( like the occurences of twin primes), is governed by clear-cut laws on the large after all.
The last Chapter is of quite differnt flavour, the so-called additive number theory. Here the author only focusses on the simplest partition function ---the unrestricted partition. However interesting phenomeon occur already at this level. The first result is Euler's pentagonal number theorem, which leads to a simple recursion formula for the partition function p(n). 3 proofs are given. The most beautiful one is no doubt a combinatorial proof due to Franklin. The third proof is through establishing the Jacobi triple product identity, which leads to lots of identites besides Euler's pentagonal number theorem. Jacobi's original proof uses his theory of theta functions, but it turns out that power series manipulaion is all that's needed.
The book ends with an indication of deeper aspect of partition theory--- Ramanujan's remarkable congrence and identities ( the simplest one being p(5m+4)=0(mod 5) ). To prove these mysterious identites, the "natural"way is to plow through the theory of modular functions, which Ramanujan had left lots more theorem ( unfortunately most without proof). However an elementary proof of one these identites is outlined in the exercises.
This book is well written, with enough exercises to balance the main text. Not bad for just an 'introduction'.