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Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics)

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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.

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.

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'.

You normally dont talk so much about readability of a book on Math, but of all the other books on number theory that I've seen, this is quite a page turner. Strikes just the right ballance between theory, proofs and examples. As mentioned somewhere in the book, one of the aims of the author is to arouse reader's interest in number theory..which this book will certainly do..especially since its main emphasis is on prime numbers.

This book is absolutely incredible. The topics covered range from some very elementary topics on the theory of certain basic arithmetic functions, to much more advanced topics such as the theory of Dirichlet L-Functions. I have never seen a clearer explanation of the characters associated with finite Abelian groups, and the L-functions associated with Dirichlet Characters, than that provided by this book. Apostol makes even the most difficult concepts seem clear and simple. As an added bonus, the end-of-chapter exercises range from moderately difficult to almost excruciatingly so (but still very fun to work on) and give the reader excellent experience in solving problems in this field. With all this said, it should be pointed out that, as another reviewer stated, this book should not be read until the reader has already had a good deal of previous exposure to number theory. I myself would recommend the book of Hardy and Wright. As a second text on number theory, and an introduction to the aspects of number theory related to function theory and analysis, I believe that Apostol's book is the best that anyone could possibly hope for.

Tom Apostol just can't write anything less than an ideal mathematics text. Here is a perfect balance between historical narrative, introductory material and more rigorous content towards the end. I will eventually buy everything he wrote.

I have just used this book, which for me culminated in studying Riemann's Famous Paper (printed in Edward's Riemann Zeta Function, you will need this or a similar book to get help through this paper). We went through every chapter in detail, and it really is an excellent book on Anaytic Number Theory, and generally as a math book.

It carefully borders the line between handholding (not too much) and terseness (just more than enough required detail) to allow the student to work at understanding the material, without feeling the books is leaving you behind (or has left out detail).

The examples are good, though I do admit that in some cases there could be a few more, and the range of difficulty in the exercises is excellent. Though I have many other books on the subject material, I rarely, if ever, needed them to help me with the material from Apostal's book.

I have one caveat, that is: I would recommend you have a book dedicated to Elementary Number Theory (I mainly used G.A.Jones, lots of exercise with solutions), for the standard err 'Elementary' stuff. Apostal coverage is good but cursory. But of course that's not what this book is about, and even though he states that a motivated and presumable talented, high schooler could do most of the book, it does require some mathematical maturity, a good understanding proofs and the really fun stuff requires an understanding of Complex Variables.

High recommended

Note: Elementary proofs of the Prime Number Theorem (of which there is an outline of one in this book) are not 'elementary', as in easy, they are usually more difficult than the analytic proofs, assuming of course you have studied Complex Analysis!

This book "Introduction to Analytic Number Theory" written by Tom Apostol, formerly from California Institute of Technology, is the best mathematical book ever written on Number Theory. Rigorous, comprehensive, elegant, well organized, it is a masterpiece that every undergraduate or graduate in mathematics should possess!

The reason I bought this book was to understand an elementary proof of the prime number theorem. Actually, it contains only an outline of an elementary proof. But the book introduces methods for the proof with awesome clarity. It must have been much greater if we could see the detailed elementary proof of the prime number theorem written by Apostol. He gives a reference to An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright for the detailed proof, but the reader may be required to do unnecessary guessing (which is believed to be good in learning math, but seems to be nothing but trouble for me) to go through it.