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Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics) Hardcover – 28 May 1998

4.7 out of 5 stars 11 customer reviews

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Product details

  • Hardcover: 356 pages
  • Publisher: Springer; 1st ed. 1976. Corr. 5th printing 1998 edition (28 May 1998)
  • Language: English
  • ISBN-10: 0387901639
  • ISBN-13: 978-0387901633
  • Product Dimensions: 16.5 x 2.2 x 23.9 cm
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (11 customer reviews)
  • Amazon Bestsellers Rank: 81,299 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Review

From the reviews:

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

“After reading Introduction to Analytic Number Theory one is left with the impression that the author, Tom M. Apostal, has pulled off some magic trick. … I must admit that I love this book. The selection of topics is excellent, the exposition is fluid, the proofs are clear and nicely structured, and every chapter contains its own set of … exercises. … this book is very readable and approachable, and it would work very nicely as a text for a second course in number theory.” (Álvaro Lozano-Robledo, The Mathematical Association of America, December, 2011)


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The theory of numbers is that branch of mathematics which deals with properties of the whole numbers, 1, 2, 3, 4, 5, . . . also called the counting numbers, or positive integers. Read the first page
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Customer Reviews

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Top Customer Reviews

Format: Hardcover
This book claims to be aimed at undergraduates but is really a postgraduate text since it assumes you already know Complex Analysis. Having said that the book is otherwise self-contained for example there is a useful introduction to Abelian Groups where these are required. No advance knowledge of number theory is assumed and the book contains an excellent exposition of Arithmetical Functions and Direchlet Multiplication, Periodic Arithmetical Functions and Gauss sums, Direchlet Series and Euler Products, culminating in an analytic proof of the Prime Number Theorem.
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Format: Hardcover
This is an OU set text for MSC level No. Theory. The book provides an excellent presentation of the subject. Its claim to accessibility by 'sophisticated' high school students is something of a stretch. This post-grad student finds the 'theorem - proof' repetition style rather terse for independent study. I have found 'Jones and Jones, Ele. No. Theory' an indispensible accompaniment. The ubiquitous use of proof by induction can be hard to follow as his style in its use seems to be all his own.
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By A Customer on 25 Jun. 2004
Format: Hardcover
From experience, this book is ideal for an undergraduate module in analytic number theory. Of course it requires knowledge of complex analysis and very basic number theory, but that is the nature of the subject and the way that most books cover it. The foundations such as Dirichlet Series and Euler Products are particularly well covered, while the Prime Number Theorem is discussed towards the end. Most topics in the more analytic side of analytic number theorem appear somewhere in the text. This book can't be beaten (except possibly by the legendary J.-P. Serre...)
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Format: Hardcover Verified Purchase
This excellent textbook explains how analysis (both real and complex) may be applied to number theory, especially to the study of the prime numbers and their distribution within the naturals. There is a lot of info about Dirichlet series and the Riemann zeta function, which is one of the most fascinating functions in pure maths. Every page is a joy to read! The ideas are clearly explained in a straight-forward way. The prose style is crystal clear and perfectly phrased.

The first few chapters deal with "elementary" ideas, such as arithmetic functions and congruences. There is also a look at group theory and Dirichlet characters, which are relevant to L series. The book ends with a look at the partition function and infinite products, which is a really interesting and beautiful topic!

There are exercises at the end of every chapter, but unfortunately there are no answers in the back, which is a shame, because the reader cannot check if his/her work is correct.

This book has been in print for about 30 years and has deservedly attained a reputation as something of a classic, in mathematical circles.

By the way, it took me a while to figure out what the diagram on the front cover is supposed to be. It's a diagram of the GCD function (greatest common divisor).
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Format: Hardcover Verified Purchase
This book gives a thorough grounding in those parts of number theory which involves analytic methods. Despite this a fairly complete treatment of elementary methods is also provided to develop firm basis for the analyic methods. The most significant results are Dirichlet Theorem on Primes in Arithmetic progressions and a thorough treatment of the Analytic Proof of the Prime Number Theorem. As a text book I have not found one that is as complete as this for a higher undergraduate or Masters level postgraduate course. The choice of material is well considered and given that no textbook can cover every aspect of a subject as vast as number theory, this book does an admirable job of providing a superb basis on which to progress to further more advanced treatise such as the classic books by Davenport (Multiplicative Number Theory) and Titchmarsh (The Theory of the Riemann Zeta Function) or Iwaniec ( Analytic Number Theory). It also provides sufficient preparation to pave the way for more advanced analytic methods such as Sieve Methods and exponential sums (see for example Cojocaru and Murty 'Introduction to Sieve methods and applications' or Harman 'Prime-Detecting Sieves')

I found that by combining this book with Burton `Elementary Number Theory' or Hardy and Wrights `Introduction to the theory of Numbers' this textbook provides an excellent introduction for anyone with an interest in Analytic Number Theory.
A highly recommended textbook.
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Format: Hardcover
Tom Apostol's works (refer to his excellent and comprehensive "Mathematical Analysis" and THAT masterpiece, "Calculus") are always marked by their elegance of presentation and lucidity in the mathematics. This book is in the same genre, which still holds its own despite many newer books on Analytic Number Theory that have come into existence since Apostol first published this over 30 years ago.

While the author claims that this is book suitable for undergraduates, much of the content requires a high degree of Pure Mathematical rigour and so would be suitable for a final year undergrad or preferably a Masters level course in the subject.

The "core" of the book is dedicated to Analytic Number Theory but it also dedicates much space and attention to other important topics. Starting with a revision of basic concepts like divisibility and the fundamental theorem of Arithmetic, the book quickly flows into the definition, inter-relationships between and application of the Arithmetic functions. The distribution of prime number is introduced early. Thereafter, following the introduction of tools like congruences, Abelian Groups and their Characters, Periodic Arithmetical Functions and Gauss Sums, the book moves into in depth exposition of Dirichlet's theorem, Dirichet Series, the Reimann Zeta function and the Analyic proof of the Prime Number Theorem. The chapter on the Prime Number Theorem also includes a useful discussion of the Riemann hypothesis. These latter chapters assume knowledge of Complex Variable Theory.

Another merit of the book are the chapters on topics like the Quadratic Reciprocity Law, Primitive Roots and Partitions. Though the book lacks worked examples, as is the case with Apostol's other books, it more than makes up for this by the numerous problems at the end of each chapter. It is a pity that hints and/or solutions are not included.
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