Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics)

Rating:
Summary: well presented, delightfully written
Review: 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 0The 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'.

Rating:
Summary: Unsurpassed SECOND text on number theory
Review: 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.

Rating:
Summary: Unsurpassed SECOND text on number theory
Review: 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.

Rating:
Summary: Excellent exercises in a clear exposition
Review: 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.

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