1.1. From the Publisher.

The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications. In order to keep the presentation reasonably self-contained, Professor Sarnak begins by developing the necessary background material in modular forms. He then considers the solution of three problems: the Ruziewicz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs: 'expander graphs' and 'Ramanujan graphs'; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares. These applications are carried out in detail. The book therefore should be accessible to a wide audience of graduate students and researchers in mathematics and computer science.

1.2. From the Preface.

These notes are an expanded version of the Wittemore Lectures given at Yale in November 1988. The material presented in the four chapters is more or less self-contained. On the other hand, in the section at the end of each chapter called 'Notes and comments,' it is assumed that the reader is familiar with more advanced and sophisticated notions from the theory of automorphic forms. Some of the material presented here overlaps with a forthcoming book, 'Discrete groups, expanding graphs and invariant measures' by A Lubotzky. The points of view, emphasis, and presentation in that book and the present notes are sufficiently different that we decided to keep the two works separate. The reader is encouraged to look at both treatments of the material.

1.3. From the Introduction.

Traditionally the theory of modular forms has been and still is, one of the most powerful tools in number theory. Recently it has also been successfully applied to resolve some long outstanding problems in seemingly unrelated fields. Our aim in this book is to describe three such applications, developing along the way the necessary methods and material from the theory of modular forms.

1.4. Review by: Solomon Friedberg.
Mathematical Reviews MR1102679 (92k:11045).

This book is concerned with the application of the theory of modular forms to the solution of three problems, each of which seems at first unrelated to the theory of modular forms. The three problems can each be reduced to the problem of estimating the size of the Fourier coefficients of modular forms, of either integral or half-integral weight.
...
This book treats in detail a remarkable range of ideas and beautiful mathematics. It is highly recommended to everyone interested in modular forms.

1.5. Review by: Robin Chapman.
Bulletin of the London Mathematical Society 24 (1) (1992), 89-90.

This book, based on a series of lectures at Yale University, is not a detailed treatise on the theory of modular forms, but rather an account of how this theory has been applied to other branches of mathematics. Three applications are given, to problems in measure theory, graph theory and number theory, respectively. The unifying thread is that all these applications use estimates on the growth of the Fourier coefficients of cusp forms. The first of the four chapters gives a terse, but readable, account of the theory of modular forms for congruence subgroups of $F = SL(2, \mathbb{Z})$. Both forms of integral and half-integral weight are treated, and the constructions of modular forms via theta functions and Eisenstein and Poincare series are dealt with. The Ramanujan conjectures on the growth of the Fourier coefficients of a cusp form of weight $k$ for a congruence subgroup of F are stated carefully, both in the case where $k$ is an even integer (proved by Deligne) and where $k$ is half of an odd integer (as yet unproved in full generality), and a proof is outlined of a slightly weaker estimate. The remaining three chapters, which are independent of one another, each deal with a separate application.
...
The author must be congratulated on producing such a well-structured book covering such a wide range of mathematics in just over a hundred pages. The style is concise, but clear, and the author avoids the trap of including too much detail, whilst succeeding in making the book almost self-contained. Only enough material is developed to cover the chosen applications, but at the end of each chapter the author includes a section called 'Notes and Comments' which indicates the links with more advanced theory. This is not a text from which one can learn the theory of modular forms, but it would complement neatly any of the standard texts on the subject. On the debit side there are a number of typographical infelicities and, more seriously, a large quantity of misprints. If, as it should, this book goes into a second edition, these blemishes must be corrected.

2.1. From the Publisher.

The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinity and related techniques from orthogonal polynomials and Fredholm determinants.

2.2. From the Introduction.

In a remarkable numerical experiment, Odlyzko found that the distribution of the (suitably normalised) spacings between successive zeroes of the Riemann zeta function is (empirically) the same as the so-called GUE measure, a certain probability measure on $\mathbb{R}$ arising in random matrix theory. His experiment was inspired by work of Montgomery, who determined the pair correlation distribution between zeroes (in a restricted range), and who noted the compatibility of what he found with the GUE prediction. Recent results of Rudnick and Sarnak are also compatible with the belief that the distribution of the spacings between zeroes, not only of the Riemann zeta function, but also of quite general automorphic $L$-functions over $\mathbb{Q}$, are all given by the GUE measure, or, as we shall say, all satisfy the Montgomery-Odlyzko Law. Unfortunately, proving this seems well beyond range of existing techniques, and we have no results to offer in this direction.

However, it is along established principle that problems which seem inaccessible in the number field case often have finite field analogues which are accessible. In this book we establish the Montgomery-Odlyzko Law for wide classes of zeta and $L$-functions over finite fields.

2.3. Review by: David Bressoud.
The American Mathematical Monthly 196 (6) (1999), 597.

There is empirical evidence that the distribution of the spacings between zeros of the zeta-function is the same as a certain probability measure from random matrix theory. This book establishes this relationship for several classes of zeta- and $L$-functions over finite fields.

2.4. Review by: D R Heath-Brown.
Bulletin of the London Mathematical Society 32 (1) (2000), 118-119.

A proof of the Riemann Hypothesis is the major missing element in the theory of the Riemann zeta-function and related zeta-functions. The Riemann Hypothesis would say that every non-trivial zero $\beta + i\gamma$ of the Riemann zeta-function lies on the line $\beta = \large\frac{1}{2}\normalsize$. However, there are many important problems for which the distribution of the imaginary parts γ of the zeros also plays a significant role. Examples include the Mertens conjecture, questions on the fine distribution of primes in short intervals, the Birch-Swinnerton-Dyer conjecture, and class-number problems.

Theoretical work of Montgomery, and numerical calculations by Odlyzko, have led to the GUE hypothesis. This states that the normalized differences

         $(\gamma' - \gamma)\Large\frac{\log\gamma}{2\pi}\normalsize$

of consecutive zeros should have the same limiting distribution as the normalized differences of the eigenvalues of a large random unitary matrix. A second question of interest concerns the distribution of the smallest zero, as one goes through a family of zeta-functions. Thus, for example, one might consider all Dirichlet $L$-functions with quadratic characters, and investigate the normalized zeros $\gamma_1 \Large\frac{\log q}{2\pi}$, where $\large\frac{1}{2}\normalsize + i\gamma_{1}$ is the smallest zero of the quadratic $L$-function to modulus $q$. Here one appears to obtain a distribution quite different from the GUE hypothesis.

In their book, Katz and Sarnak investigate the corresponding questions for the zeta-functions of varieties over finite fields. In this situation, they are actually able to prove distribution laws for the zeros of zeta-functions of 'almost all' curves of large genus, in the limit as both the genus and the size of the finite field tend to infinity. For the spacings between consecutive zeros, they establish the GUE hypothesis. However, the situation for the distribution of the smallest zero, as one runs through a family of varieties, is rather more interesting, in that the result obtained depends on the associated monodromy group. The authors therefore propose that there should be some type of monodromy group associated to a family of global zeta-functions, and dependent on the form of the functional equation. This would explain very well the differing behaviour of $\gamma_{1}$ for various families of zeta-functions. ...

To a large extent, this book is an extended research paper, and much of the material is new. The first eight chapters give a detailed treatment of the spacing of eigenvalues for groups of large matrices. The book then goes on to give the necessary results on varieties over finite fields. This leads up to the equidistribution of certain Frobenius conjugacy classes in the geometric monodromy group. Finally, the two aspects of the theory are combined to handle the distribution of differences of consecutive zeros, and of the smallest zero, of the zeta-function attached to the variety.

This book is not for the faint-hearted. The material is wide ranging and difficult. However, for research workers interested in the Riemann Hypothesis, or in the arithmetic of varieties over finite fields, this work has important messages which may help to shape our thinking on fundamental issues on the nature of zeta-functions.

2.5. Review by: Philippe G Michel.
Mathematical Reviews MR1659828 (2000b:11070).

This book is fascinating in many aspects: First, its rigorous, systematic and accessible exposition of the subject makes it a bright landmark at the crossroads of arithmetic and mathematical physics; no doubt it will become a basic reference in random matrix theory. Second, it offers its reader a bouquet of beautiful new results but also leaves the door open to many challenging conjectures. The general GUE-type models for zeros of $L$-functions provided by the density hypothesis and the philosophy stemming from it reveal a brand new, stimulating aspect of the theory of $L$-functions (see for example the beautiful expository article by the authors [Zeroes of zeta functions and symmetry (1999)]): this opens a handful of possible new approaches to making accurate predictions (as in physics), to interpreting puzzling coincidences, and finally to proving theorems. This book will constitute, for those working in the field of $L$-functions, a major source of inspiration for the next decades.

3.1. From the Publisher.

This text is a self contained treatment of expander graphs and in particular their explicit construction. Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering. The reader needs only a background in elementary algebra, analysis and combinatorics; the authors supply the necessary background from graph theory, number theory, group theory and representation theory. Thus the text can be used as a brief introduction to these subjects and their synthesis in modern mathematics.

3.2. From the Preface.

These notes are intended for a general mathematical audience. In particular, we have in mind that they could be used as a course for undergraduates. They contain an explicit construction of highly connected but sparse graphs known as expander graphs. Besides their interest in combinatorics and graph theory, these graphs have applications to computer science and engineering. Our aim has been to give a self-contained treatment. Thus, the relevant background material in graph theory, number theory, group theory, and representation theory is presented. The text can be used as a brief introduction to these modern subjects as well as an example of how such topics are synthesised in modern mathematics. Prerequisites include linear algebra together with elementary algebra, analysis, and combinatorics.

3.3. Review by: Thomas J Pfaff.
Mathematical Association of America (1 April 2004).
https://maa.org/press/maa-reviews/elementary-number-theory-group-theory-and-ramanujan-graphs

Loosely speaking, a family of $k$-regular graphs is called a family of expanders if the size of the vertex sets goes to infinity, while the graphs maintain good connectedness. Such graphs are important in engineering applications such as network designs, complexity theory, derandomization, coding theory and cryptography. Elementary Number Theory, Group Theory, and Ramanujan Graphs is devoted to constructing the Ramanujan graphs which are a family of expanders. Moreover, these graphs provide an explicit example of an infinite family of graphs with large girth and large chromatic number. The large girth and chromatic number problem was originally solved by Erdös using the probabilistic method, but this does not provide a construction of such graphs.

The book covers a considerable amount of mathematical ground in order to construct and prove the results about the Ramanujan graphs: linear algebra (eigenvalues and spectral gaps), number theory (sums of two and four squares and quadratic reciprocity), and group theory (general linear groups and representation theory of finite groups). Along the way the reader will also see operators between $L^{2}$ spaces, Chebyshev polynomials, the ring of quaternions, metabelian groups, and Cayley graphs. The fact that all these topics are used to prove graph theory results is what makes this book so interesting. The book is broken up into four chapters covering graph theory, number theory, group theory, and the Ramanujan graphs. In fact, the first three chapters can be read independently and each one is interesting.

The preface of the book claims that this book could be used for an undergraduate course. Based on the topics above I will let you decide if it is appropriate for an undergraduate course at your institution. I think that it would be difficult to use at most undergraduate colleges even as a senior capstone type course. On the other hand, any of the first three chapters could be used for an independent study course with undergraduates. The book would make a nice elective course for graduate students since it pulls so many topics together. If you are looking for a book for a faculty seminar that isn't too discipline-specific, this would be a good choice.

Overall, the book is a well written and stimulating book. My only complaint is that the book doesn't actually give any examples of the applications. It does give references for the applications, but a couple of pages devoted to enlightening the reader about the applications would have been worthwhile.

3.4. Review by: Thomas R Shemanske.
Mathematical Reviews MR1989434 (2004f:11001).

The purpose of this book is to describe the family of Ramanujan graphs introduced and studied by Lubotzky, Phillips, Sarnak and Margulis, and to give an elementary and self-contained exposition which demonstrates that the graphs have (the vast majority of) the desired properties.

The book is aimed at a mathematical audience who has seen a first course in abstract algebra, and perhaps a little analysis and combinatorics, and who is game for a fast-track introduction to selected topics in combinatorics, elementary number theory, and the linear representation theory of finite groups. It would make a great text for an honours or senior seminar, showing how elegantly many different areas of mathematics come together to solve a very concrete problem of broad interest and application.

After a brief overview, the text delves into graph theory, discussing issues of girth, chromatic number and spectral gap. From graph theory, it moves to number theory, discussing representations of integers as sums of two and four squares, taking advantage of arithmetic in the Gaussian integers and integral quaternions. It also develops the formulas for the number of representations in terms of divisor functions.

Following the number theory, the authors take an in-depth look at the group $PSL_{2}(q)$. They quickly establish the simplicity of the group for $q ≥ 4$, and then move towards bounding the degrees of irreducible representations of $PSL_{2}(q), q ≥ 5$ a prime. This seems the most demanding material for those with a light algebra background, as the authors quickly develop the notions of linear representations, characters, and tensor products.

From here they return to graph theory, characterising the Ramanujan graphs as Cayley graphs associated to $PSL_{2}(q)$. They establish the connectedness and regularity of the graphs. While they cannot (via elementary means) establish that the graphs are Ramanujan, they do establish that the graphs are all expanders.