1.1. From the Preface.

I was captivated by a group of enthusiastic Spanish mathematicians whose desire for cultivating modern number theory I enjoyed recently during two memorable events, the first at the summer school in Santander, 1992, and the second while visiting the Universidad Autónoma in Madrid in June 1993. These notes are an expanded version of a series of eleven lectures I delivered in Madrid. They are more than a survey of favourite topics, since proofs are given for all important results. However, there is a lot of basic material which should have been included for completeness, but was not because of time and space limitations. Instead, to make a comprehensive exposition we focus on issues closely related to the spectral aspects of automorphic forms (as opposed to the arithmetical aspects, to which I intend to return on another occasion].

Primarily, the lectures are addressed to advanced graduate students. I hope the student will get inspiration for his own adventures in the field. This is a goal which Professor Antonio Córdoba has a vision of pursuing throughout the new volumes to be published by the Revista Matemática Ibercamericana. I am pleased to contribute to part of his plan.

1.2. Review by: Solomon Friedberg.
Mathematical Reviews MR1325466 (96f:11078).

Let $H$ denote the complex upper half plane and let $\Gamma$ be a discrete finite-volume subgroup, such as a congruence subgroup. Then one may study the space of $L^{2}$-functions on $\Gamma H$. This book offers an introduction to the study of this space, that is, to the development of the spectral theory of $GL(2)$ automorphic forms. The material and exposition are well-suited for second-year or higher graduate students.

The book begins with a quick review of harmonic analysis on the plane, followed by a discussion of harmonic analysis on the hyperbolic plane $H$. Fuchsian groups are then defined and discussed. After these introductory chapters the central objects of study, (non-holomorphic) modular forms, are defined. As an important example (for $\Gamma$ not co-compact), Eisenstein series are presented and their Fourier expansions are computed.

The next chapters, comprising about 50 pages, develop the spectral theorem in detail. In particular, the analytic continuation of the Eisenstein series is proved, using Selberg's approach based upon the Fredholm theory of integral equations. ...

The material on the spectral theorem is followed by a brief chapter concerning estimates for the Fourier coefficients of Maass forms. Included are estimates for the Fourier coefficients on average with respect to the spectrum, which are proved by use of the spectral theory previously developed. The author then turns to the Bruggeman-Kuznetsov formula, which is obtained by applying the spectral theorem to Poincaré series.

Next comes the Selberg trace formula. This formula establishes a quantitative connection between the spectrum of the Laplacian and the geometry of the Riemann surface $\Gamma \backslash H$. The formula is developed, and applications are presented: the Selberg zeta-function, an asymptotic law for the length of closed geodesics on $\Gamma \backslash H$, and Weyl's law. The "big problem of small eigenvalues" is also discussed.

The concluding chapters cover hyperbolic lattice-point problems, which are studied by evaluating the automorphic kernel $K$ for a properly chosen function $k$, and recent joint work of the author and P Sarnak on $L^{∞}$-norms for cusp forms on the modular group.

There are two appendices, the first reviewing facts from classical analysis (self-adjoint operators, Hilbert-Schmidt operators, Fredholm operators, the Green function of a differential equation) and the second concerning special functions.

One would think that we are at page 400, but in fact the references start on page 233. This clear and comprehensive book concerning the spectral theory of $GL(2)$ automorphic forms belongs on many a bookshelf.

2.1. From the Publisher.

The book is based on the notes from the graduate course given by the author at Rutgers University in the fall of 1994 and the spring of 1995. The main goal of the book is to acquaint the reader with various perspectives of the theory of automorphic forms. In addition to detailed and often nonstandard exposition of familiar topics of the theory, particular attention is paid to such subjects as theta-functions and representations by quadratic forms.

2.2. From the Preface.

Automorphic forms are present in almost every area of modern number theory. They also appear in other areas of mathematics and in physics. I have lectured on these topics many times at Rutgers University with only slight overlapping of content, and I still have new material to teach that is important. It is indeed a vast territory which cannot be grasped by any one person. While research publications on automorphic forms are rapidly increasing in quantity and quality, the demand for textbooks, particularly on the graduate level, is also growing. There are fine books on this subject, but still more are needed, especially those which favour analytic methods.

The present book is based on my lecture notes (almost verbatim except for Section 5.5) from a graduate course which I delivered in the Fall of 1994 and in the Spring of 1995 at Rutgers. The course was formulated, as the title implies, to acquaint our new students with the subject matter from various perspectives. Thus I have not followed direct or traditional paths, but rather I have frequently ventured into areas where different ideas and methods mix and interact. To cover a lot in a limited time, some material is necessarily presented as a survey. For example, the numerous connections of automorphic forms with $L$-functions of number fields are discussed in Chapter 12 without detail. However we do provide complete proofs of the most basic results in the earlier sections.

An experienced reader will find some of our arguments to be nonstandard. It would be pointless to argue which approach is better, since our choice was made simply for the purpose of showing different possibilities. For example, our presentation of the theory of Hecke operators in Chapter 6 is completed for primitive characters quickly by establishing the multiplicity-one principle using Gauss and Ramanujan sums instead of lengthy considerations of inner products. Of course, this is only a special case (the whole space is spanned by newforms), but it is an important case.

We pay great attention to detail in the subjects of theta functions and representations by quadratic forms (Chapters 10 and 11) because these are not sufficiently covered in textbooks, despite having a long history of research.

Because the original notes were written as the course was progressing, inevitably some redundancy has occurred. Nevertheless we have decided not to eliminate this redundancy, because it offers the option of selective reading. For example, our account of the Shimura-Taniyama conjecture for special curves (the congruent number curves) is self-contained in Chapter 8, even though one could instead appeal to the later chapters on general theta functions.

Sergei Gelfand, Peter Sarnak and others have convinced me that these lecture notes might be useful for a large number of graduate students, and they have urged me to publish them. I would like to thank them for their encouragement.

2.3. Contents.

Preface.

Chapter 0. Introduction.

Chapter 1. The Classical Modular Forms.
1.1. Periodic functions.
1.2. Elliptic functions.
1.3. Modular functions.
1.4 The Fourier expansion of Eisenstein series.
1.5. The modular group.
1.6. The linear space of modular forms.

Chapter 2. Automorphic Forms in General.
2.1. The hyperbolic plane.
2.2. The classification of motions.
2.3. Discrete groups - Fuchsian groups.
2.4. Congruence groups.
2.5. Double coset decomposition.
2.6. Multiplier systems.
2.7. Automorphic forms.
2.8. The eta-function and the theta-function.

Chapter 3. The Eisenstein and Poincaré Series.
3.1. General Poincaré series.
3.2. Fourier expansion of Poincaré series.
3.3. The Hilbert space of cusp forms.

Chapter 4. Kloosterman Sums.
4.1. General Kloosterman sums.
4.2. Kloosterman sums for congruence groups.
4.3. the classical Kloosterman sums.
4.4. Power-moments of Kloosterman sums.
4.5. Sums of Kloosterman sums.
4.6. The Salié sums.

Chapter 5. Bounds for the Fourier Coefficients of Cusp Forms.
5.1. General estimates.
5.2. Estimates by Kloosterman sums.
5.3. Coefficients of cusp forms with theta multiplier.
5.4. Linear forms in Fourier coefficients of cusp forms.
5.5 Spectral analysis of the diagonal symbol.

Chapter 6. Hecke Operators.
6.1. Introduction.
6.2. Hecke operators $T_{n}$.
6.3. The Hecke operators on periodic functions.
6.4. The Hecke operators for the modular group.
6.5. The Hecke operators with a character.
6.6. An overview of newforms.
6.7. Hecke eigencupsforms for a primitive character.
6.8. Final remarks.

Chapter 7. Automorphic $L$-functions.
7.1. Introduction.
7.2. The Hecke $L$-functions.
7.3. Twisting automorphic forms and $L$-functions.
7.4. Converse theorems.

Chapter 8. Cusp Forms Associated with Elliptic Curves.
8.1. The Hasse-Weil $L$-function.
8.2. Elliptic curves $E_{r}$.
8.3. Computing $\lambda (p)$.
8.4. A Hecke Grossencharacter.
8.5. A theta series.
8.6. The automorphy of $f$.

Chapter 9. Spherical Functions.
9.1. Positive definite quadratic forms.
9.2. Space spherical functions.
9.3. The spherical functions reconsidered.
9.4. Harmonic analysis on the sphere.

Chapter 10. Theta Functions.
10.1. Introduction.
10.2. An inversion formula.
10.3. The congruent theta functions.
10.4. The automorphy of theta functions.
10.5. The standard theta function.

Chapter 11. Representations of Quadratic Forms.
11.1. Introduction.
11.2. Siegel's mass formula.
11.3. Representations by Eisenstein series and cusp forms.
11.4. The circle method of Kloosterman.
11.5. The singular series.
11.6. Equidistribution of integral points on ellipsoids.

Chapter 12. Automorphic Forms Associated with Number Fields.
12.1. Automorphic forms attached to Dirichlet $L$-functions.
12.2. Hecke $L$-functions with Grossencharacters.
12.3. Automorphic forms associated with quadratic fields.
12.4. Class group $L$-functions reconsidered.
12.5. $L$-functions of genus characters.
12.6. Automorphic forms of weight one.

Chapter 13. Convolution $L$-functions.
13.1. Introduction.
13.2. Rankin-Selberg integrals.
13.3. Selberg's theory of Eisenstein series.
13.4. Statement of general results.
13.5. The scattering matrix for $\Gamma _{0}(N)$.
13.6. Functional equations for the convolution $L$-functions.
13.7. Metaplectic Eisenstein series.
13.8. Symmetric power $L$-functions.

2.4. Review by: Balakrishnan Ramakrishnan.
Mathematical Reviews MR1474964 (98e:11051).

The theory of automorphic forms is becoming an integral part of modern number theory. There are several textbooks as well as advanced books written on this subject and still there is a demand for books because of the vastness of this subject. The present book by the author is meant for advanced researchers in this field.

The book is organised into thirteen chapters along with a very brief introduction to the subject.
...
This book differs from other texts in many aspects. For example, one can see from the contents that the author discusses many important topics in the theory of automorphic forms which are rarely seen in the textbooks available on the subject. Another aspect is the presentation of the proofs, which is also unusual, and this may give the reader a different flavour of the subject. The author could have included more references in the bibliography, and this is the only weak point in this book. Graduate students will certainly benefit from this book.

2.5. Review by: Jonathan D Rogawski.
Bulletin of the American Mathematical Society 35 (3) (1998), 253-263.

... one can take the point of view that automorphic forms are primarily of interest because of the concrete analytic information they give us about classical problems. In this optic, functoriality is a tool rather than an end in itself, and a wide range of other methods from analytic number theory play an equally important role. This is the approach of Iwaniec in Topics in classical automorphic forms. Like the other two books under review, Iwaniec devotes several chapters to standard background material: the modular group, Eisenstein series, Hecke operators, $L$-functions, etc. However, the main focus is on two problems: (1) estimating the size of the Fourier coefficients of a modular form and (2) representing integers by quadratic forms.
...
Iwaniec treats a number of other topics, though sometimes without detailed proofs: newforms, Weil's converse theorem, automorphic forms attached to Hecke characters and elliptic curves, Eisenstein series, and the Rankin-Selberg method. Although analytically demanding in parts, the exposition is clear, and helpful explanatory comments are included throughout. If I have one small complaint, it is that the author did not include an overview section similar to the "Concluding Remarks" section in [A Borel, Automorphic forms on $SL_{2}(\mathbb{R})$]. Such a section could have described the state of the field and provided the reader with a useful guide of where to go next. Nevertheless, this is an excellent place to begin the study of the analytic approach to modular forms.

3.1. From the Publisher.

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style.

The first edition of this book was an underground classic, both as a textbook and as a respected source for results, ideas, and references. The book's reputation sparked a growing interest in the mathematical community to bring it back into print. The American Mathematical Society has answered that call with the publication of this second edition.

In the book, Iwaniec treats the spectral theory of automorphic forms as the study of the space $𝐿^{2}(𝐻 \Gamma$), where $𝐻$ is the upper half-plane and $\Gamma$ is a discrete subgroup of volume-preserving transformations of $𝐻$. He combines various techniques from analytic number theory. Among the topics discussed are Eisenstein series, estimates for Fourier coefficients of automorphic forms, the theory of Kloosterman sums, the Selberg trace formula, and the theory of small eigenvalues.

Henryk Iwaniec was awarded the 2002 AMS Cole Prize for his fundamental contributions to analytic number theory. Also available from the American Mathematical Society by H Iwaniec is Topics in Classical Automorphic Forms, Volume 17 in the Graduate Studies in Mathematics series.

The book is designed for graduate students and researchers working in analytic number theory.

3.2. From the Preface.

Since the initial publication by the Revista Matemática Iberoamericana I have used this book in my graduate courses at Rutgers several times. I have heard from my colleagues that they also had used it in their teaching. On several occasions I have been told that the book would be a good source of ideas, results and references for graduate students, if it were easier to obtain. The American Mathematical Society suggested publishing an edition, which would increase its availability. Sergei Gelfand should be given special credit for his persistence during the past five years to convince me of the need for a second publication. I am also very indebted to the editors of the Revista for the original edition and its distribution and for the release of the copyright to me, so this revised edition could be realised.

I eliminated all the misprints and mistakes which were kindly called to my attention by several people. I also changed some words and slightly expanded some arguments for better clarity. In a few places I mentioned recent results. I did not try to include all of the best achievements, since this had not been my intention in the first edition either. Such a task would require several volumes, and that could discourage beginners. I hope that a subject as great as the analytic theory of automorphic forms will eventually be presented more substantially, perhaps by several authors. Until that happens, I am recommending that newcomers read the recent survey articles and new books which I have listed at the end of the regular references.

3.3. Contents.

Introduction
Chapter 0. Harmonic analysis on the Euclidean plane.
Chapter 1. Harmonic analysis on the hyperbolic plane.
Chapter 2. Fuchsian groups.
Chapter 3. Automorphic forms.
Chapter 4. The spectral theorem. Discrete part.
Chapter 5. The automorphic Green function.
Chapter 6. Analytic continuation of the Eisenstein series.
Chapter 7. The spectral theorem. Continuous part.
Chapter 8. Estimates for the Fourier coefficients of Maass forms.
Chapter 9. Spectral theory of Kloosterman sums.
Chapter 10. The trace formula.
Chapter 11. The distribution of eigenvalues.
Chapter 12. Hyperbolic lattice-point problems.
Chapter 13. Spectral bounds for cusp forms.
Appendix A. Classical analysis.
Appendix B. Special functions.

4.1. From the Publisher.

Analytic Number Theory distinguishes itself by the variety of tools it uses to establish results. One of the primary attractions of this theory is its vast diversity of concepts and methods. The main goals of this book are to show the scope of the theory, both in classical and modern directions, and to exhibit its wealth and prospects, beautiful theorems, and powerful techniques.

The book is written with graduate students in mind, and the authors nicely balance clarity, completeness, and generality. The exercises in each section serve dual purposes, some intended to improve readers' understanding of the subject and others providing additional information. Formal prerequisites for the major part of the book do not go beyond calculus, complex analysis, integration, and Fourier series and integrals. In later chapters automorphic forms become important, with much of the necessary information about them included in two survey chapters.

4.2. From the Preface.

This book shows the scope of analytic number theory both in classical and modern directions. There are no division lines; in fact our intent is to demonstrate, particularly for newcomers, the fascinating countless interrelations. Of course, our picture of analytic number theory is by no means complete, but we tried to frame the material into a portrait of a reasonable size, yet providing a self-contained presentation.

We were writing this book in a period of time during and after teaching courses and working with graduate students in Rutgers University, Bordeaux University and Courant Institute. We thank these institutions for providing conditions for both of us to work together. We shared ideas on what this book should be about with many of our colleagues, who gave us critical suggestions. Among them we would like to mention Etienne Fouvry, John Friedlander, Philippe Michel and Peter Sarnak. During a long process of typing and preparation of this book for publication, we received stimulating encouragement and technical advice from Sergei Gelfand, for all of his help we express our gratitude. Carol Hamer helped to polish some of our English phrases while her little boys tried to destroy the TeX files without success. We thank them all for the output.

4.3. Contents.

Chapter 1. Arithmetic functions.
Chapter 2. Elementary theory of prime numbers.
Chapter 3. Characters.
Chapter 4. Summation formulas.
Chapter 5. Classical analytic theory of $L$-functions.
Chapter 6. Elementary sieve methods.
Chapter 7. Bilinear forms and the large sieve.
Chapter 8. Exponential sums.
Chapter 9. The Dirichlet polynomials.
Chapter 10. Zero density estimates.
Chapter 11. Sums over finite fields.
Chapter 12. Character sums.
Chapter 13. Sums over primes.
Chapter 14. Holomorphic modular forms.
Chapter 15. Spectral theory of automorphic forms.
Chapter 16. Sums of Kloosterman sums.
Chapter 17. Primes in arithmetic progressions.
Chapter 18. The least prime in an arithmetic progression.
Chapter 19. The Goldbach problem.
Chapter 20. The circle method.
Chapter 21. Equidistribution.
Chapter 22. Imaginary quadratic fields.
Chapter 23. Effective bounds for the class number.
Chapter 24. The critical zeros of the Riemann zeta function.
Chapter 25. The spacing of the zeros of the Riemann zeta-function.
Chapter 26. Central values of $L$-functions.

4.4. Review by: Glyn Harman.
Bulletin of the London Mathematical Society 37 (2005), 316-317.

What constitutes 'Analytic number theory' as a subject, and sets it apart from other areas of number theory, is not entirely clear-cut. There is a grey area between elementary and analytic number theory, and any field of number theory is likely to interact with every other part. In its broadest definition, analytic number theory is the application of analytic methods (real, complex or harmonic) to problems involving integers or integer-like sets (the ring of integers in an algebraic number field, for example). In particular, the distribution of primes (or prime-like elements) plays a central role in the subject. The book under review has an enormous scope, covering large areas of modern-day analytic number theory. The classical application of real, complex and Fourier analysis is present, but also the modern use of ideas from modular forms and spectral analysis is described. In addition, interactions with algebraic geometry are given, and applications are made to algebraic number-fields.
...
Among the fundamental subjects covered in some way in this book, we mention: the prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, characters (Dirichlet and Hecke), summation formulae, functional equations, zeros of $L$-functions and zero-free regions (including automorphic $L$-functions), sieve methods, the use of bilinear forms, exponential sums, Dirichlet polynomials, mollifiers, sums over finite fields (including the Weil bound by Stepanov's elementary method), sums over primes, Kloosterman sums, the Bombieri–Vinogradov theorem, Linnik's theorem on the least prime in an arithmetic progression, additive problems, equidistribution, imaginary quadratic fields, zeros of $\zeta(s)$ on Re $s = \large\frac{1}{2}\normalsize$, and connections with random matrix theory. It would have been impossible for the authors to have covered all these subjects in depth. They have therefore opted to tackle some aspects in detail, while giving sketches and surveys for others (with references to more specialised papers or books). The advantage of such a wide coverage is that interactions can be seen between apparently different fields within the subject. Also, the earlier, 'classical' results can often be viewed as specific examples of a modern, more general, approach.

The exposition is generally of high quality, although of necessity some sections consist only of brief sketches. The authors emphasise the ideas behind the proofs, and take the view that the mathematics should be 'beautiful'. One drawback of their approach, however, is that many mathematicians will need to read through this book several times if they are to understand all the comments of the authors. The subject cannot be viewed as a straight-line development, and so there are many references forward in the text, which may be comprehensible only on a second or third reading. This is but a small price to pay, given the encyclopaedic nature of this volume.

Who should buy this book? Certainly, every researcher in analytic number theory should have this tome on their shelf, for no other work covers such a wide range of subjects. Nor do many other authors have the depth of insight of the present writers. They are experts in this area, and have produced an important reference work that should also be of value to researchers in other branches of number theory. The authors state in their introduction: "Formal prerequisites for much of the book are rather slight." So, does this work supersede other books that one might recommend to research students starting out. In my opinion, a student would benefit most from this book if they already had a background in number theory. For many students, a mastery of the classical theory would also be most helpful. This book by Iwaniec and Kowalski would then be an excellent introduction to 21st-century analytic number theory.

4.5. Review by: Kannan Soundararajan.
Mathematical Reviews MR2061214 (2005h:11005).

This outstanding book makes accessible a substantial portion of analytic number theory. The book leads the reader from classical techniques and results in analytic number theory to the frontiers of research. At least as far as multiplicative number theory is concerned, the scope of this book is almost encyclopaedic and the book will be both useful for students, and a valuable reference for researchers.

The first four chapters of the book give a quick account of arithmetical functions, characters, elementary results on prime number, and various useful summation formulas (including the important Poisson summation formula). Chapter 5 is the first of many distinctive and important chapters of this book. Here the authors develop the classical analytic theory of $L$-functions, including discussions of zero-free regions and the Riemann hypothesis, and applications to the prime number theorem and prime number theorem for arithmetic progressions. Importantly, the authors develop the basic analytic theory for a wide class of $L$-functions: Dirichlet $L$-functions, $L$-functions of number fields, classical (i.e. $GL(2)$) automorphic $L$-functions, general automorphic $L$-functions (here the analytic properties are described concretely, and in detail) and Artin $L$-functions. This chapter is a mine of information which was previously available only in the research literature, and is a good example of how the book will be useful in providing perspective for the student while at the same time providing a convenient reference for the researcher.

The next two chapters return to more basic material, and develop sieve methods. The combinatorial sieve, Selberg's sieve, and the large sieve are all discussed, together with more recent analogues of the large sieve (such as Deshouillers and Iwaniec's work on the large sieve for cusp forms). Complete proofs are given for the basic results, and the more advanced results are sketched and set in context. Chapter 8 develops the classical methods of Weyl, van der Corput and Vinogradov for estimating exponential sums, and indicates the consequences for zero-free regions of $\zeta(s)$. Chapters 9 and 10 develop zero density results which, in applications, can often substitute for the Riemann hypothesis.

Chapter 11 gives a very useful discussion of bounds complete exponential and character sums, which are very useful in analytic number theory. In particular, this chapter gives an account of Stepanov's elementary method and uses this to prove the Weil bound for Kloosterman sums. Then a brief, but informative, survey of $l$-adic cohomology is given with a view to "give the reader enough knowledge to make at least a preliminary analysis of any exponential sum he or she may encounter in analytic number theory."
...
,,, this book covers a vast amount of analytic number theory. It will be an invaluable resource for researchers in the area. It is perhaps too advanced to serve as a first introduction to the subject, but students with some familiarity of analytic number theory will find much here that is deep and beautiful.

4.6. Review by: Alexandru Zaharescu.
Bulletin of the American Mathematical Society 43 (2) (2006), 273-278.

The book under review is an important comprehensive account of modern ana- lytic number theory. It is written in a relaxed, personal style with graduate students in mind, but there is plenty in it for the expert. The book is often witty, and in a long and unusually useful introduction there is a panoramic survey of the subject in which the authors set out their reasons for the choices of topics they have made. The 26 chapters give a masterly introduction to many of the most exciting areas of contemporary research. Of course, there is not room for everything, but I think it is fair to say that the authors' objectives are still the major themes of analytic number theory. They demonstrate how the huge diversity of analytic methods have made an impact on the field. They write with authority and show their mastery of all the material on every page. One might say that they do for modern theory what Landau did for it in the first half of the last century, no mean achievement given all that has been accomplished since Landau's time.

The book opens with five introductory chapters, one connected to the other and meant to be read consecutively, on the classic topics of arithmetic functions, elementary prime number theory, characters, summation formulas, and the analytic theory of $L$-functions. These chapters are suitable for supplementary reading for a beginning graduate course in analytic number theory. The discussion in the later chapters is invigorating and serious. It leads the reader through an abundance of highly nontrivial and sophisticated technical details - for example, on exponential sums, Kloosterman sums, equidistribution theory, averages of Fourier coefficients, and correlation of zeros of the Riemann zeta-function - that are ideal for graduate topics courses that stimulate further study in specialised monographs and research articles. One of the delightful features of this book is that inside each section are interesting exercises that provide additional information about the subject and that provoke and encourage one to make one's own discoveries in the subject at hand. The mathematical prerequisites call for some acquaintance with the fundamentals of elementary number theory, advanced calculus involving inequalities, complex analysis and integration, and abstract algebra.

Euler is credited with being the first mathematician to use analytical arguments for the purpose of investigating properties of integers, by constructing generating power series, although, at least in principle, ideas relating analysis with number theory can be traced before him. It is worth mentioning that, before Euler, a number of great mathematicians had attempted to establish exact formulas for the computation of the transcendental numbers $\pi$ and $e$. In retrospect, their power series expansions and continued fraction expansions for these numbers show a clear and conscientious effort of using tools from analysis in order to better understand the nature of numbers. Euler pushed further the use of methods from analysis to questions about number theory. His use of the zeta-function and the corresponding product over primes not only proved the existence of infinitely many primes, which was known to Euclid, but it also established a quantitative relation, namely that the sum of inverses of primes is divergent. This paved the way for the appearance of asymptotic formulas in number theory, and constitutes the beginning of analytic number theory.
...
There are several classical and notable books that introduce new researchers to the basic results, methods, and terminology of analytic number theory and, at the same time, serve the dual purpose of textbooks for graduate students in many different fields of mathematics. Considering that the literature in the field has grown prodigiously in the last hundred years, none to date has been written so ambitiously and on such a grand scale as the book under review. I predict that the book will have a major impact on research and on aspiring young mathematicians.

5.1. From the Publisher.

This is a comprehensive and up-to-date treatment of sieve methods. The theory of the sieve is developed thoroughly with complete and accessible proofs of the basic theorems. Included is a wide range of applications, both to traditional questions such as those concerning primes, and to areas previously unexplored by sieve methods, such as elliptic curves, points on cubic surfaces and quantum ergodicity. New proofs are given also of some of the central theorems of analytic number theory; these proofs emphasise and take advantage of the applicability of sieve ideas. The book contains numerous comments which provide the reader with insight into the workings of the subject, both as to what the sieve can do and what it cannot do. The authors reveal recent developments by which the parity barrier can be breached, exposing golden nuggets of the subject, previously inaccessible. The variety in the topics covered and in the levels of difficulty encountered makes this a work of value to novices and experts alike, both as an educational tool and a basic reference.

5.2. From the Preface.

What possesses us to write a book on sieve methods? Hopefully we have something to say that is worth listening to and, if we didn't write it, maybe a little something would be lost.

Many young people are drawn to mathematics and frequently what is most attractive to them are problems which are related to the most elementary objects, the integers, the squares, and the primes. The sieve method, itself an elementary idea two thousand years old, offers, even to the high school teenager, a natural approach within her own compass. Many of our colleagues can relate to this experience, as do we. Just as the student during his development acquires more sophisticated tools and tastes, so has the subject of sieve methods itself; and, whereas it is the elementary ideas that offer the initial attraction, it is the ability to infuse these with the more advanced tools that brings joy in later life. One of our goals is to transmit some of these joys to new investigators.

Eratosthenes, about 300 BC, is the first to be mentioned. His idea, illustrated in Chapter I, is essentially an algorithm for tabulating primes. Following a very long gap, the subject was taken up by A Legendre who gave a formula (Section 1.1) for $\pi(x)$. This was the beginning of the principle of inclusion-exclusion. Legendre's formula, however, is of very limited practical value and the ideas which were needed to turn it into a useful instrument were initiated by V Brun in 1915. He developed his methods over the following decade before leaving the topic. We discuss his ideas in Chapter 6. Such a late placement should alert the reader that we do not follow a chronological order or sieve developments in this book.

The combinatorial complexity of Brun's most advanced works inhibited later researchers so that the subject did not receive the huge initial impetus that the power of his ideas merited.

Brun's results were improved considerably by A Buchstab in the late 1930's using an iterative scheme, the basis of which has remained valuable to this day and which is d cussed in several chapters, especially Chapter 11.

A new impetus came to the subject with the ideas of A Selberg in the late 1940's (discussed in Chapter 7). His upper-bound sieve, which, following Selberg, we call the $\Lambda ^{2}$ sieve, is, in particular, very elegant in comparison with the earlier Brun sieves. It is also considerably stronger in some respects. although subsequent developments have shown that the strongest results are obtained by keeping both the $\Lambda ^{2}$-sieve and the combinatorial sieve in mind.

More recent progress has involved many new ideas and many names. Not only has the theory of the sieve apparatus grown but it has developed enhancements which permit the implementation of sophisticated results from harmonic analysis, from exponential sums, from arithmetic geometry, and from automorphic forms. The reachable targets have expanded, not only front almost-primes to primes, but also to questions of greater variety, for example to solutions of Diophantine equations. We hope the contents of the book will speak for themselves.

Let's take a quick run-through of the chapters. Although we follow roughly the order of development of the subject there are significant detours. Thus, for example, Bombieri who appears in Chapter 3 is thought to be somewhat younger than Eratosthenes who appears in Chapter 4. The reader we hope has sufficient mathematical maturity not to be troubled by this.
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We had a number of goals in writing this book. Foremost among them, we would like to encourage young people to study the subject. In doing so it would be very helpful, in case they do not already have the relevant background, to have at hand some books on analytic number theory. An inexperienced reader may find it helpful to also consult some classic texts in sieve theory. For experienced readers, we also recommend the article of Selberg as a companion to the more theoretical aspects of our book.

We do intend the book also for more experienced readers. especially from other parts of number theory and mathematics, whether to muster the subject or to learn those parts they need for application. We made an effort to make the book a handy reference for theorems and techniques. Experience shows, however, that frequently the theorems available are not precisely in the form that is wanted for a particular application. So, it is important to be in command of the ideas behind the theorems in order to perform the necessary adjustments, sometimes small and sometimes not so small.

A word of warning. A lot of ground is covered and the shortest route is not always chosen. The book is intentionally not written in a linear order. Some topics require tools developed later, some topics are discussed before earlier discoveries and some topics are more difficult than others which are treated later. Don't get discouraged! Although we hope that very few readers will choose to take a minimal route, an exceedingly bare-bones introduction to the subject can be at tempted by means of Sections 1.1-1.3, Sections 5.1-5.8, Sections 6.1-6.5 and 6.9, and finally, Sections 7.1, 7.4-7.6 and 7.8-7.12.

On the other hand, we did not intend to make the book encyclopaedic. The subject of sieve methods has undergone tremendous growth in the past few decades, especially when it comes to applications, many of them highly interesting. Already on their own, Erdos and his collaborators are responsible for dozens of these. Reluctant choices needed to be made, including omission of some personal favourites of the authors. Inevitably, there are instruments which we do not allow to perform in our opera. Nevertheless, we hope that there will be readers who find plenty of interesting points and will devote their time to the further development of the subject. Not least, we hope we have succeeded in conveying to many an appreciation of the music that is the sieve.

5.3. Review by: D R Heath-Brown.
Mathematical Reviews MR2647984 (2011d:11227).

This book is a comprehensive account of the classical theory of sieve methods in number theory, and of their applications. Written by the two leading authorities on the subject, it contains a wealth of insights as well as a string of new results. Other good books in this area include those by H Halberstam and H-E Richert, G Greaves and G Harman. In comparison to these, Opera de cribro takes a wider view of sieves, and gives a broader range of applications. It shows clearly the connections with number theory in general. None of these books is an easy read, and some students may find the exposition of technicalities more to their taste in the work by Halberstam and Richert. Others will appreciate Harman's detailed treatment of asymptotic formulae for sifting functions, with their applications to prime numbers. However, for an overview of "what sieves are all about", this book by Friedlander and Iwaniec is surely the best.
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The topics covered here are remarkably wide-ranging, as are the connections which the authors make. Indeed this book is not just a volume on sieves, but rather a treatise on analytic number theory more generally. As such it is recommended to everyone with an interest in the subject.

5.4. Review by: Frank Thorne.
Bulletin of the American Mathematical Society 50 (2) (2012), 359-366.

How many prime numbers are there?

This simple question has inspired the subject of analytic number theory. The sharpest known results use the theory of complex variables, and of the Riemann zeta function in particular.

However, our original question is purely elementary (if not easy!), inviting study by elementary methods. Generally speaking, these elementary methods are known as sieve methods, and they run from the very simple to the extraordinarily sophisticated. Sieve methods are valued not only for their aesthetic value as an elementary approach, but also for their flexibility: they have proved useful in studying a wide variety of questions related to the primes, in some cases where zeta function techniques are not applicable.

Reading the Latin title of the second book under review, one might wonder, Is the study of sieves really 2,000 years old, dating back to ancient Rome? In fact ... the subject goes back further, to ancient Greece.
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... the door [is] open for a modern treatment of sieve methods, written for the aspiring expert, which connects classical work to ongoing progress. This is brilliantly accomplished by Opera de cribro. On the back cover Enrico Bombieri calls the book "a true masterpiece," and your reviewer found no cause to disagree.

Caveat emptor, however: the book is not for the faint of heart. This can be seen, for example, in their treatment of the sieve of Eratosthenes.
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This book, then, is recommended for those who have already read an introductory book on analytic number theory. The authors freely use techniques such as Mellin integration and Poisson summation, which are familiar to the seasoned analytic number theorist but which could catch the novice by surprise. Occasionally the authors appeal to more sophisticated results from the subject, for which it would be handy to also have Iwaniec and Kowalski's book Analytic number theory at hand.

Although the casual reader might prefer to look elsewhere, the serious and prepared reader will recognise this book as a goldmine. Among many other places, this can be seen in Chapter 7.2, "Comments on the $\Lambda ^{2}$-Sieve". Having presented the main theorem of the Selberg sieve in Chapter 7.1, the authors now offer five full pages of discussion of the result. The formulas for the sieve coefficients are (unavoidably) complicated, and so this section reads like a burst of fresh air. The serious student no longer has any excuse to treat the Selberg sieve as a mysterious black box, as Friedlander and Iwaniec take great pains to provide not only the details but also the motivation.

The authors discuss technical details such as "composition of sieves" early in the book, and they do the difficult work of establishing the fundamental theorems of sieve theory in the middle (roughly Chapters 7-11). The payoff is apparent later in the book, where the authors present a dizzying variety of variations and applications - far more than in any book on sieves that I have seen.
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Opera de cribro does an impressive job of presenting sieve methods as a genuine theory, where the technical underpinnings are well motivated and for the most part encapsulated, and one really can just "introduce an upper-bound sieve."
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This exciting and innovative book will introduce the reader to a fascinating area of contemporary research, which is very much intertwined with the rest of analytic number theory, and which has a promising future - even if it does not prove the twin prime conjecture. The authors are sure to be rewarded for their efforts by the sight of ratty, worn-out copies of their book in the offices of a generation of analytic number theorists.

5.5. Review by: Enrico Bombieri.
Institute for Advanced Study.

This monograph represents the state of the art both in respect of coverage of the general methods and in respect of the actual applications to interesting problems.

A unique feature of this monograph is how the authors take great pains to explain the fundamental ideas behind the proofs and to show how to approach a question in a correct fashion. So, this book is not just another monograph useful for consultation; rather, it is a teaching instrument of great value both for the specialist and the beginner in the field.

The authors must be congratulated for this exceptional monograph, the first of its kind for depth of content as well as for the effort made to explain the 'why' and not limiting themselves to the 'how to'. This is a true masterpiece that will prove to be indispensable to the serious researcher for many years to come.

6.1. From the Publisher.

The Riemann zeta function was introduced by L Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics.

The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.

Undergraduate and graduate students and research mathematicians interested in number theory and complex analysis.

6.2. From the Preface.

The Riemann zeta function $\zeta(s)$ in the real variable $s$ was introduced by L Euler (1737) in connection with questions about the distribution of prime numbers. Later B Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. He revealed a dual correspondence between the primes and the complex zeros of $\zeta(s)$, which started a theory to be developed by the greatest minds in mathematics. Riemann was able to provide proofs of his most fundamental observations, except for one, which asserts that all the non-trivial zeros of $\zeta(s)$ are on the line Re $s = \large\frac{1}{2}\normalsize$. This is the famous Riemann Hypothesis - one of the most important unsolved problems in modern mathematics.

These lecture notes cover closely the material which I presented to graduate students at Rutgers in the fall of 2012. The theory of the Riemann zeta function has expanded in different directions over the past 150 years; however my goal was limited to showing only a few classical results on the distribution of the zeros. These results include the Riemann memoir (1859), the density theorem of F Carson (l920) about the zeros off the critical line, and the estimates of G H Hardy - J E Littlewood (1921) for the number of zeros on the critical line. Then, in Part 2 of these lectures, I present in full detail the result of N Levinson (1974), which asserts that more than one third of the zeros are critical (lie on the line Re $s = \large\frac{1}{2}\normalsize$). My approach had frequent detours so that students could learn different techniques with interesting features. For instance, I followed the stronger construction invented by J B Conrey (1983), because it reveals clearly the essence of Levinson's ideas.

After establishing the principal inequality of the Levinson-Conrey method, it remains to evaluate asymptotically the second power-moment of a relevant Dirichlet polynomial, which is built out of derivatives of the zeta function and its mollifier. This task was carried out differently than by the traditional arguments and in greater generality than it was needed. The main term coming from the contribution of the diagonal terms fits with results in sieve theory and can be useful elsewhere.

I am pleased to express my deep appreciation to Pedro Henrjque Pontes, who actively participated in the course and be gave valuable mathematical comments, which improved my presentation. He also helped significantly in editing these notes in addition to typing them.

6.3. Contents.

Preface

Part 1. Classical Topics.

Chapter I. Panorama of Arithmetic Functions.
Chapter 2. The Euler-Maclaurin Formula.
Chapter 3. Chebyshev Prime Seeds.
Chapter 4. Elementary Prime Number Theorem.
Chapter 5. The Riemann Memoir.
Chapter 6. The Analytic Continuation.
Chapter 7. The Functional Equation.
Chapter 8. The Product Formula over the zeros.
Chapter 9. The Asymptotic Formula for $N(T)$.
Chapter 10. The Asymptotic Formula for $\psi(x)$.
Chapter 11. The Zero-free Region and the PNT.
Chapter 12. Approximate Functional Equations.
Chapter 13. The Dirichlet Polynomials.
Chapter 14. Zeros off the Critical Line.
Chapter 15. Zeros on the Critical Line.

Part 2. The Critical Zeros after Levinson.

Chapter 16. Introduction.
Chapter 17. Detecting Critical Zeros.
Chapter 18. Conrey's Construction.
Chapter 19. The Argument Variations.
Chapter 20. Attaching a Mollifier.
Chapter 21. The Littlewood Lemma.
Chapter 22. The Principal Inequality.
Chapter 23. Positive Proportion of the Critical Zeros.
Chapter 24 . The First Moment of Dirichlet Polynomials.
Chapter 25. The Second Moment of Dirichlet Polynomials.
Chapter 26. The Diagonal Terms.
Chapter 27. The Off-diagonal Terms.
Chapter 28. Conclusion.
Chapter 29. Computations and the Optimal Mollifier.

Appendix A. Smooth Bump Functions.
Appendix B. The Gamma Function.

6.4. Review by: Sidney West Graham.
Mathematical Reviews MR3241276.

This slim volume is a notable addition to the literature on the Riemann zeta-function. Instead of taking the encyclopaedic approach of the books by E C Titchmarsh [The theory of the Riemann zeta-function] or A Ivić [The Riemann zeta-function], the author focuses on an important theorem of N Levinson, which states that, asymptotically, at least 34% of the zeros of the Riemann zeta-function have real part $\large\frac{1}{2}\normalsize$. This is, of course, an approximation to the Riemann Hypothesis, which states that all non-trivial zeros have real part $\large\frac{1}{2}\normalsize$.

Part I of the book consists of 15 chapters, and it begins with basic results on arithmetic functions, Euler-Maclaurin summation, and Chebyshev's prime number estimates. The author then proceeds to develop the basic elements of zeta-function theory that are necessary for Levinson's proof. Part I ends with the 1921 Hardy-Littlewood result that the number of zeros on the critical line Re$(s) = \large\frac{1}{2}\normalsize$ up to height $T$ is >> $T$.

Part II consists of 14 chapters and gives a complete proof of Levinson's theorem. The exposition incorporates some simplifications due to J B Conrey and numerous technical improvements due to the author. The pace of the exposition is brisk, but there are frequent side comments about interesting technical features. The book originated in lecture notes given to graduate students at Rutgers in 2012. It is accessible to a reader with a first course in complex analysis; however, some prior experience with analytic number theory would be helpful.

6.5. Review by: Brian Conrey.
Bulletin of the American Mathematical Society 53 (3) (2016), 507-512.

I hope that Hilbert hasn't jinxed us with his prediction of the future. He purportedly said during a lecture that a proof of the Riemann Hypothesis was forthcoming within a few years, that some in the room might live to see a proof of Fermat's Last Theorem, but that the proof of the irrationality of $2^{√2}$ was centuries away. But, Gelfond and Schneider proved the latter a few years later and Wiles proved Fermat's Last Theorem within that century. So one hopes that Hilbert did not have it exactly backwards!

Analytic number theorists like to claim that the Riemann Hypothesis is the most important unsolved problem in mathematics. With Fermat's Last Theorem and the Poincaré Conjecture out of the way, it is not so difficult to make that argument convincingly! But also the mounting evidence in its favour and the ubiquity of $L$- functions that also have Riemann Hypotheses convince us of the centrality of this problem. The fact that it has been more than 150 years since Riemann posed his problem adds to the argument. Henryk Iwaniec's new book on the Riemann zeta function gives a fascinating perspective on the subject that will be relished by beginners and experts alike. Like his other beautiful books, Iwaniec gives us a modern treatment of something of great interest to contemporary mathematicians.

I wish I could say that the proof of the Riemann Hypothesis is just around the corner, but I'm afraid that would be extremely misleading. But what I can say is that there is more evidence than ever that it is true. The burning question about the Riemann Hypothesis is "Why is it seemingly so hard?" The best answer I can give is that the Riemann zeta function is a complicated function; more complicated than the other functions we generally encounter.
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Iwaniec's lovely new book is a splendid window on what many consider the greatest mathematics problem of all time!

6.6. Review by: Alberto Perelli.
Hardy-Ramanujan Journal 39 (2016), 63-64.

These lecture notes arise from a course given by the author to graduate students at Rutgers University in the fall of 2012. The book is divided in two parts, having different aims and levels.

The first part introduces the basic theory of the Riemann zeta function $\zeta (s)$ as well as some of its applications to the distribution of prime numbers. Actually, $\zeta (s)$ was first introduced for real values of $s$ by Euler, who used it to give a new proof of the existence of infinitely many primes. In contrast with Euclid's purely arithmetical reasoning, Euler's proof combines analytic and arithmetical arguments, and may be regarded as the starting point of analytic number theory.

Part 1 begins with four chapters reviewing the approach to the study of the distribution of primes by elementary methods, where complex function theory is not used. In particular, Chapter 4 presents an interesting short elementary proof of the Prime Number Theorem, and discusses related questions. Chapters 5-11 contain the basic theory of the Riemann zeta function, starting with a brief description of Riemann's 1859 original paper and ending with the proof of the classical zero-free region and the Prime Number Theorem. Chapters 12-15, which conclude Part 1, enter the finer theory of $\zeta (s)$. The first two chapters present several forms of the approximate functional equation and some basic facts about Dirichlet polynomials. Such results are then applied, in the next two chapters, to the study of the horizontal distribution of the zeta zeros. Indeed, the famous Riemann Hypothesis (1859) asserts that all complex zeros of ζ (s) lie on the critical line Re $s = \large\frac{1}{2}\normalsize$. This would completely solve the problem of the horizontal distribution, and many approximations to the Riemann Hypothesis have been obtained by several authors, starting with the famous theorems of Hardy and of Bohr & Landau published in 1914. Indeed, Hardy's theorem shows that there are infinitely many zeta zeros on the critical line, while the Bohr-Landau theorem asserts that almost all zeros lie in an arbitrarily small strip around such a line. In these directions, Chapter 14 contains an account of Carlson's density theorem, the first result giving sharp quantitative upper bounds for the number of zeros of $\zeta (s)$ off the critical line, and Chapter 15 presents a well known achievement of Hardy & Littlewood on the zeros on the critical line. Here, the approximate functional equation is used to show that there are >> $T$ zeros on the critical line up to height $T$. We refer to Sankaranarayanan's survey, in this volume, of Ramachandra's contributions to the Riemann zeta function for several related problems and results.

Part 2 is devoted to the more recent work on the zeros of $\zeta (s)$ on the critical line, leading to sharp improvements of Hardy-Littlewood's theorem mentioned above. The first result showing that a positive proportion of the zeros lie on the critical line is a famous theorem due to Selberg in 1942, obtained by an ingenious refinement of the Hardy-Littlewood approach involving several important innovations. ...
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I wish to stress once more the richness of Iwaniec's book; the standard material is offered in a concise but instructive way, and from the more advanced part one can feel the deep insight of the author. This is therefore a highly welcome and a very useful addition to the literature on the Riemann zeta function and related topics.