1.1. From the Preface.

This is a tract dealing with Fourier transforms and some topics naturally connected with them, and although the material included is familiar, if not classical, there is not much of a duplication with other books in the field.

Acknowledgement of thanks is due from Bochner to the Office of Naval Research, and from Chandrasekharan to the Institute for Advanced Study.

Princeton University and the Institute for Advanced Study. November 1948.

1.2. Review by: J L B Cooper.
The Mathematical Gazette 35 (312) (1951), 140-141.

A number of topics covering the parts of the real variable theory of Fourier transforms important for applications and the related General Transforms are dealt with in this book. It begins with the theory of Fourier transforms of functions on $L (-∞, ∞)$, treating their convergence and summability theory, with some relatively elementary uniqueness theorems. There is a thorough discussion of summability by a very general class of kernels, and applications of these results to give theorems about ordinary convergence. The second chapter deals with functions of several variables, containing a number of contributions which the authors have made to the subject, particularly the theory of summability by means of radial kernels. Chapter III discusses $L_{p}$ spaces, giving some account of the general theory of these spaces and of Banach spaces in general, but little of the theory of Fourier transforms in these spaces. Chapter IV develops the theory for $L_{2}$ spaces, both in one and in many variables, very thoroughly. Chapter V, after a preliminary account of general unitary transforms on $L^{2}(0, ∞)$ induced by integral transforms, deals with Watson transforms, proving a number of interesting results not hitherto available in the books on the subject. The last chapter deals with Tauberian theorems, whose study is begun in the first the treatment is, fundamentally, that of Wiener, but there are many new ideas, and the proofs and results differ from those of Wiener and his collaborators.

The attitude of the book is modern, making consistent use of the theory of linear spaces. This gives particular liveliness to the treatment of the Watson transforms. The section on multiple Fourier transforms fills a very definite gap in the literature, made the more glaring by the many applications of these transforms based on no rigorous foundation.

The application of Fourier transforms to the solution of differential equations is little discussed, presumably because it is so fully treated elsewhere, and also because the purely real-variable approach of the book would make strong restrictions on the size of solutions at infinity necessary: this is illustrated by the two cases discussed, the heat conduction equation and the potential equation which arise in connection with Gaussian and Abelian summability respectively.

As a whole the book is very readable, and could be studied profitably by anyone with a knowledge of the Lebesgue integral. The one exception is the last chapter, where the analysis is rather heavy. The production of the book, by photography from typescript, is well and clearly done. ...

1.3. Review by: Harry Pollard.
Mathematical Reviews MR0031582 (11,173d).

This treatment is concerned largely with the Fourier transform in $L$ and $L^{2}$ both for functions of one and of several variables. Chapters I-IV contain standard material such as the inversion formulas, Abel and Gauss summability, an account of the $L^{p}$ spaces (although the transform in $L^{p}$ is not treated), and applications to simple boundary value problems. A theorem which the authors consider worthwhile to investigate further is Theorem 9 of Chapter I: let $f(x)$ belong to $L$ and let its Fourier transform $\phi(x)$ be nonnegative; if $f(x)$ is bounded in a neighbourhood of the origin then $\phi(x)$ is also in $L$. A weaker version of this theorem is used by Lévy (1948), and is attributed by him to Loève. Chapter V is concerned with the general unitary transformation of $L^{2}(0, ∞)$ and with Watson transforms. It is based on the well-known paper of Bochner (1934). To the bibliography of this chapter one ought to add the work of Kacmarz (1933). The final Chapter VI, on general Tauberian theorems, is motivated by the work of Karamata and Wiener, but the treatment follows the lines of a paper of Bochner (1934).

1.4. Review by: I E Segal.
Bulletin of the American Mathematical Society 56 (6) (1950), 526-528.

This is a very readable introduction to the craft of the authors, and as such fills a very real need. The subject matter serves to a large extent as a springboard for the presentation of interesting techniques, viewpoints, and concepts, and is treated with great deftness and remarkable continuity. There is a good deal of explanatory and motivating material, and altogether the book is a very appropriate one for study by an apprentice to the guild of semi-classical analysts.

The book is apparently not intended for reference use, nor is its subject matter and development the sort that are best adapted to the needs of a mathematician with a merely general interest in the subject, or to the needs of a theoretical physicist. The topics treated are interesting but the basis of their selection seems to have been aesthetic and subjective, rather than a function of their relative significance within the general framework of mathematics. Among the important topics not treated are Fourier transforms in the complex domain, Fourier-Stieltjes transforms, and generalised harmonic analysis; from a purely informational standpoint the book is hardly a well-rounded survey of Fourier analysis. The proofs and approaches employed in the book are technically elementary, but often somewhat intricate and delicate, and demand either a certain sophistication or special attentiveness of the reader. The very considerable smoothness, coherency, and local clarity of the book seem to be achieved by a unity of mathematical viewpoint and technique, as well as by niceness of style, rather than by logical organisation.

Fourier transforms of functions in $L_{1}$ and $L_{2}$ on the real line and in euclidean $n$-space are treated along familiar lines, but with much intrinsically interesting illustrative material. In particular, transforms of derivatives, and Gauss and Abel summability, are presented in some detail. Special but valuable topics more briefly treated include transforms of radial functions, closure of translations of functions in $L_{2}$, and bounded transformations on $L_{2}$ commuting with translations. All these developments take up the first four chapters of the book, and there are two additional chapters, concerning more specialised matters, along lines reflecting some of the particular interests of the authors. The fifth chapter gives an integro-differential representation for the most general unitary operator on $L_{2}$, from which a study of the Watson transform evolves. The sixth chapter treats aspects of Tauberian theorems from a combination of viewpoints due originally to Karamata and Wiener. These last two chapters both have a certain roundness and charm almost, which make relatively complex subjects seem quite approachable. This feature may be related to the circumstance that each of these chapters is apparently based on a paper by Bochner.

There is also a short treatment of Banach spaces, along with a discussion of the $L_{p}$ spaces for general $p$ (the Fourier transforms of functions in those spaces are not considered). None of the deeper theorems about Banach spaces is covered, and on the whole the "abstract" viewpoint is avoided. The outlook of the book is indicated by the following quotation (pp. 211-212): "... theorem 7... has the disadvantage (if disadvantage it is) of being peculiar to the euclidean setup ..." (reviewer's italics). The book contains essentially no orientation of the results with respect to the theories of harmonic analysis on topological groups, Banach algebras, or operators on Hilbert space. (One of the very few comments on the operator-theoretic background of a result is unsound, - the operators described on page 215 as having simple spectrum can have nearly arbitrary spectral multiplicities.) It opens with a definition of Fourier transform on the line, and no explanation is even given of why this particular transform, rather than some other, is the object of such intensive study. The fact that there exists a general theory of transforms or locally compact abelian groups, which establishes such basic results as the uniqueness theorem for $L_{1}$ transforms, the closure theorems for translations of functions in $L_{1}$ and $L_{2}$, and the generalised Plancherel theorem, in forms which apply to Fourier integrals and series on $n$ dimensional euclidean space, as well as to almost periodic functions expansions into Walsh-Rademacher functions, functions on finite groups, and so forth, is devoid of any significant recognition.

Thus the book is intensive rather than extensive in scope, and strength of logical organisation has been somewhat sacrificed for elegance of technique and mathematical style. It is a really valuable addition to the literature, and substantial numbers of students should find it both pleasant and instructive.

2.1. From the Preface.

This book deals with the theory of 'typical means' and its applications to Dirichlet series and Fourier series. More than forty years have now passed since 'typical means' were first introduced by M Riesz for the summation of divergent series, and quite an extensive theory has developed during this period. We have attempted here to give a systematic account of this development. Readers of our account will hardly need to be told how much we owe to the Cambridge tract by Hardy and Riesz on the general theory of Dirichlet series.

2.2. Review by: Szolem Mandelbrojt.
Mathematical Reviews MR0055458 (14,1077c).

Hardy and Riesz's little book [The General Theory of Dirichlet Series, Cambridge, 1915], out of print for many years, was devoted to Marcel Riesz's typical means and their applications to the Dirichlet series. The present monograph, by taking up the subject again and systematically and concisely summarising the essential progress made since then in this vast theory, which is dotted with results of varying importance, happily fills a gap. Those who fondly remember their first reading of Hardy and Riesz's excellent Cambridge Tract, as well as younger readers just beginning to explore the field, will be grateful to the authors.

2.3. Review by: T.S.N.
Current Science 22 (6) (1953), 187.

This book inaugurates the series of Monographs on mathematics and physics launched by the Tata Institute of Fundamental Research, Bombay, mainly for the use of research workers. It comprises some important work on typical means hitherto available only in research periodicals and forms a valuable supplement to the Cambridge Tract by Hardy and Riesz on the general theory of Dirichlet series. Written by two eminent mathematicians of this country whose own contributions in this field are quite significant, it is bound to acquire a distinguished place in the literature of summability.

An extensive theory has developed since the introduction of 'typical means' by Marcel Riesz more than forty years ago for the summation of divergent series. The book under review is a laudable attempt to give a systematic account of this development. It proceeds in four chapters, the first two of which are devoted to an exposition of the theory of typical means. Chapter I introduces the notions of summability by Rietz's typical means and of absolute Riesz summability due to Obrechkoff, and establishes, for the former, the first theorem of consistency, a limitation theorem and some Tauberian theorems. The analogue of the first theorem of consistency for absolute Riesz summability concludes the chapter.

Chapter II discusses the second theorem of consistency for Riesz summability with a classification of the different versions of this theorem due to Hardy, Zygmund and Hirst. Chapter III deals with the applications of the theory to Dirichlet series, giving Abelian theorems as well as some new converse theorems on the abscissae of summabilities, theorems on the behaviour of a Dirichlet series on the line of summability, Tauberian theorems for Dirichlet summability and many new theorems on the Dirichlet product of summable series. The last chapter is concerned with the application of Riesz's means to the study of spherical summation of multiple Fourier series largely due to Bochner and Chandrasekharan, and it is based on a formula of Bochner which may be regarded as a generalisation of the classical integral of Fejér. It commences with the notion of spherical summation of multiple Fourier series and series derived therefrom, leading to applications to summations over lattice points in k dimensions and to a solution of the problems of ordinary and absolute summabilities discussed in the case of single Fourier series by Hardy, Littlewood and others.

The printing and get-up of the book are excellent. The notes at the end of each chapter with references brought up-to-date compensate for the lack of an index and a bibliography.

2.4. Review by: H G Eggleston.
The Mathematical Gazette 38 (323) (1954), 61-62.

As the first mathematical monograph of a new series of publications by the Tata Institute this book, intended primarily for use by professional mathematicians actively engaged on research, sets a high standard by its accuracy and clear presentation.

The scope of many convergence arguments in analysis can often be extended by an appropriate interpretation of any divergent series (or integrals) that occur. Each such interpretation gives rise to a method of summability. Although there are many interrelations between different methods of summability it is usually the case that there is one method, or one type of method, that is of outstanding importance in any particular problem. Typical Means or Riesz Means are a method devised to handle the convergence problems that arise in connection with Dirichlet Series.

The first two chapters of this monograph are devoted to the definitions and fundamental properties of typical means. The third and longest chapter deals with their application to Dirichlet Series and the fourth chapter with the method of spherical summability applied to multiple Fourier Series.

The book is written with great care and is full of results which have previously only been available in widely scattered articles in various periodicals. Some results given here are new and all the theorems are proved with as much generality as is possible. Throughout the book there is a coherence of method and of the structure of the results obtained that enable the reader to grasp the main features of the theory comparatively readily. But it is in the nature of the subject that such a book as this cannot be easy to read.

The authors are to be congratulated on having performed a necessary and useful service in producing a book which fills a very obvious gap in mathematical literature. The information contained in this monograph is completely up-to-date and there is no doubt that it will serve as the starting point for many future researches.

At the conclusion of each chapter are "Notes" which refer to original papers and contain other relevant comments. The collection of these notes at the end of each chapter is not so convenient for the reader as their insertion at the proper place in the text would have been. It may be remarked from these notes that this is a branch of mathematics in which mathematicians in Britain have made fundamental contributions and that the subject owes a great deal to the work of G H Hardy, L S Bosanquet and B Kuttner.

2.5. Review by: B Kuttner.
Bulletin of the American Mathematical Society 60 (1) (1954), 85-88.

This book, which is the first of a new series of monographs to be published under the auspices of the Tata Institute of Fundamental Research, Bombay, is concerned entirely with the theory and applications of Riesz summability. In view of the fact that Hardy's Divergent series devoted only a little space to Riesz summability, there was room for another book dealing more fully with this particular method. While reasonably complete, the book is not exhaustive; indeed, an account of everything that is known on the subject would be impossible in a book of this size. However, a useful series of notes at the end of each chapter contains numerous references, and also the statement of certain theorems for which room was not found in the main text. The book collects in a convenient form much material which has hitherto been available only in the original papers, and it should prove of great use to anyone who wants to work in this particular field. It is a pity that its usefulness should be diminished by the errors which occur in it.
...

3.1..From the Preface.

The aim of these lectures is to provide an introduction to the theory of the Riemann Zeta-function for students who might later want to do research on the subject. The Prime Number Theorem, Hardy's theorem on the Zeros of $\zeta(s)$, and Hamburger's theorem are the principal results proved here. The exposition is self-contained, and required a preliminary knowledge of only the elements of function theory.

4.1. From the Preface.

This book has grown out of a course of lectures I have given at the Eidgenossische Technische Hochschule, Zurich. Notes of those lectures, prepared for the most part by assistants, have appeared in German. This book follows the same general plan as those notes, though in style, and in text (for instance, Chapters III, V, VIII), and in attention to detail, it is rather different. Its purpose is to introduce the non-specialist to some of the fundamental results in the theory of numbers, to show how analytical methods of proof fit into the theory, and to prepare the ground for a subsequent inquiry into deeper questions. It is published in this series because of the interest evinced by Professor Beno Eckmann.

I have to acknowledge my indebtedness to Professor Carl Ludwig Siegel, who has read the book, both in manuscript and in print, and made a number of valuable criticisms and suggestions. Professor Raghavan Narasimhan has helped me, time and again, with illuminating comments. Dr Harold Diamond has read the proofs, and helped me to remove obscurities. I have to thank them all.

4.2. Review by: John Hunter.
Proceedings of the Edinburgh Mathematical Society 17 (1) (1970), 112.

This is volume 148 in the famous "Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen" series. As one would expect, in the short space of 130 medium sized pages of text, a book containing 11 chapters cannot enter deeply into many topics. Nevertheless the book is written in a style that avoids any sense of packing; for example, three proofs are given of the unique factorisation theorem, each involving the introduction of different sets of necessary basic ideas. The fact that the book is not of the type with exercises within and at the end of each chapter gives the author considerable freedom in the choice of material and in the choice of proofs. For example, after a fairly standard introduction to quadratic residues, the law of quadratic reciprocity is proved from the reciprocity property for generalised Gaussian sums. This of course assumes the fair amount of necessary complex variable theory and could not be given until a fairly late stage of an Honours course. The two main results proved are Dirichlet's theorem on primes in an arithmetical progression and the prime number theorem. For the first of these a fairly standard use of Dirichlet's $L$-functions is employed, all the necessary work on characters and Dirichlet series being described, and, for the second, a proof using the Wiener-Ikehara theorem is given in terms of Chebyshev's function 4, all the necessary work on arithmetical functions and estimates on the distribution of prime numbers being given in earlier chapters. Two other chapters contain work on Weyl's theorems on uniform distribution and Kronecker's theorem and Minkowski's theorem on lattice points in convex sets. While the claim that the book is written for the non-specialist may not be entirely justified, nevertheless it is written in a most attractive style with interesting notes at the end and printed with the usual high standard associated with this series.

4.3. Review by: L Kuipers.
Mathematical Reviews MR0249348 (40 #2593).

This has grown out of a course given by the author at the Eidgenössische Technische Hochschule, Zürich. The first version of the book appeared as a publication in the series, Lecture Notes in Mathematics [Einführung in die analytische Zahlentheorie, Springer, Berlin, 1966]. The purpose of the book is to introduce the non-specialist to some of the fundamental results in the theory of numbers and to show how analytical methods of proof fit into the theory. There are eleven chapters, treating consecutively: The unique factorisation theorem, Congruences, Rational approximations and Hurwitz's theorem, Quadratic residues and the representation of a number as a sum of four squares, The law of quadratic reciprocity, Arithmetical functions and lattice points, Chebyshev's theorem on the distribution of prime numbers, Weyl's theorems on uniform distribution and Kronecker's theorem, Minkowski's theorem on lattice points in convex sets, Dirichlet's theorem on primes in an arithmetical progression, The prime number theorem.

4.4. Review by: Clas-Olof Selenius.
Nordisk Matematisk Tidskrift 19 (1/2) (1971), 40-41.

The book is actually just a translation of the author's earlier compendium "Einführung in die analytische Zahlentheorie" in the Lecture notes series. Some reworking is said to have been done, but the deviations are infinitesimal. The translation is nevertheless not without justification: Lecture notes are not as commercially available as the "yellow series".

Chandrasekharan's style is simple and clear. The presentation meets fairly high pedagogical standards. It is noticeably verbal, in contrast to the daunting formalistic style that is now all the rage, especially in the USA. Fortunately, the sections are not numbered decimally and the typography is quite good. The beginner thus has the necessary initial prerequisites to be able to read the book without frustration.

The introductory chapters are devoted to divisibility and congruences. Then the author goes into rational approximation of irrational numbers. Here he prefers Farey fractions to continued fractions, as he does not need to go further than Hurwitz's theorem. In chapters 4 and 5 quadratic residues and the reciprocity theorem are treated, which the author derives from the reciprocity of generalised Gaussian sums. Thus the presentation has been analysed.

The next chapter is devoted to various arithmetic functions: the lattice point function $r(n)$, the factor function $d(n)$, the functions $\sigma(n)$, $\mu(n)$ and $\phi(n)$, and the Mangoldt function. A neat result is: there is a positive constant $C$ such that $C < \sigma(n)\phi(n)/n^{2} < 1$, then $n ≥ 2$.

Chebyshev's theorem becomes the gateway to the prime distribution in Chapter 7. It is proven traditionally (as, for example, in Nagell's book), while Bertrand's postulate, i.e. the theorem that between a natural number and its double there is always a prime, is proven in connection with Pillai. The prime number theorem itself is postponed to Chapter 11.

Then, uniform distribution mod 1 and the classical theorems of Weyl are treated. For Kronecker's famous theorem, which according to Hardy is one of the theorems that show that what is not impossible will happen at some point, no matter how improbable, are given by Bohr's proof, and he follows Hardy's book almost verbatim (Hardy also gave two "non-analytical" proofs of the theorem).

Minkowski's theorem on lattice points in convex sets (in $\mathbb{R}^{n}$) is proved and applied in Chapter 9. The proof method is Siegel's and it is based on a formula for the measure of a finite, measurable, convex, symmetric set with no lattice points other than the origin.

Thus the author has arrived at the final goal: the famous parade numbers Dirichlet's theorem on prime numbers in an arithmetic series and the prime number theorem. The verifications in the proof of the former are nicely done and the proof is divided into tasty portions: recapitulation of Euler's identity, characters, the introduction of Dirichlet series, Landau's theorem, the $L$-functions and finally the main theorem. The proof of the prime number theorem starts from the proof that the Riemann zeta function $\zeta(1+it)$ does not vanish and is based on the Wiener-Ikehara theorem.

The book ends with notes, which include historical notes, references and additions. In the note to Dirichlet's theorem, it is noted that the author counts on the following so-called elementary proofs: Mertens' in 1897, Selberg's in 1950 and Zassenhaus' in 1949, one year before Selberg. Erdös is not mentioned in connection with the prime number theorem.

4.5. Review by: H M Stark.
Bulletin of the American Mathematical Society 77 (6) (1971), 943-957.

We begin this review with a discussion of Chandrasekharan's Introduction to analytic number theory, which is a translation with some slight revisions of the author's Einführung in die analytische Zahlentheorie (Springer lecture notes series number 29). This book presupposes the usual knowledge of functions of a complex variable (i.e. Cauchy's theorem) but virtually no knowledge of number theory. Indeed, the book begins with the unique factorisation theorem and in the early chapters moves through (among other things) congruences, the law of quadratic reciprocity and several standard arithmetical functions. The later chapters include Weyl's theorems on uniform distribution, Minkowski's convex body theorem, Dirichlet's theorem on primes in arithmetic progressions and the prime number theorem (via the Wiener Ikahara method).

Several of the theorems in the book are proved by means of convex variables and Fourier series (one of the nicest of these being Siegel's proof of Minkowski's theorem). Still there is no circle method, no functional equations, nor even any Dirichlet series extended past their half plane of convergence. Indeed, a Dirichlet series with a complex argument is not even mentioned until p. 106 (24 pages before the end of the text). In conclusion, I would say that the book is interesting and well written but it is definitely not an introduction to analytic number theory.

4.6. Review by Allen Stenger.
Mathematical Association of America Reviews (4 August 2017).

This is a concise sampler of analytic number theory, introducing problems in a number of different areas of number theory and showing how they can be attacked by analytic methods. It doesn't go into a great deal of depth on any problem. The prerequisites are a moderate knowledge of complex analysis. The necessary number theory is developed from scratch. In some ways it resembles D J Newman's Analytic Number Theory, although that book deals with more difficult problems and is not really an introduction.

The exposition owes a great deal to Hardy & Wright's An Introduction to the Theory of Numbers and to Landau's Elementary Number Theory, except for the last two chapters, on Dirichlet's theorem on primes in arithmetic progressions and on the Prime Number Theorem. The former uses a complex-variables proof depending on Landau's theorem that a positive-term Dirichlet series has a singularly on the abscissa of convergence. The latter is a nice exposition of the Wiener-Ikehara proof that requires no knowledge of Fourier integrals. Other topics covered include Bertrand's postulate, quadratic reciprocity (through Gauss sums), average orders of arithmetic functions, equidistribution, and a good bit on the geometry of numbers.

The big weakness of this book, apart from its high price, is that it's too shallow and doesn't lead you anywhere. For example, it does have a very nice proof of the prime number theorem, but you don't find out that there is a whole industry devoted to improving our knowledge of the Riemann zeta function or how it connects to prime number theory. A good alternative to this book is Apostol's Introduction to Analytic Number Theory. It also starts at the beginning, covers the roughly the same topics, but goes into much greater depth. Apostol also has very good exercises, while the present book has none.

5.1. From the Preface.

The plan of this book had its inception in a course of lectures on arithmetical functions given by me in the summer of 1964 at the Forschungsinstitut für Mathematik of the Swiss Federal Institute of Technology, Zürich, at the invitation of Professor Beno Eckmann. My Introduction to Analytic Number Theory has appeared in the meanwhile, and this book may be looked upon as a sequel. It presupposes only a modicum of acquaintance with analysis and number theory.

The arithmetical functions considered here are those associated with the distribution of prime numbers, as well as the partition function and the divisor function. Some of the problems posed by their asymptotic behaviour form the theme. They afford a glimpse of the variety of analytical methods used in the theory, and of the variety of problems that await solution.

I owe a debt of gratitude to Professor Carl Ludwig Siegel, who has read the book in manuscript and given me the benefit of his criticism. I have improved the text in several places in response to his comments. I must thank Professor Raghavan Narasimhan for many stimulating discussions, and Mr Henri Joris for the valuable assistance he has given me in checking the manuscript and correcting the proofs.

5.2. Review by: L Kuipers.
Mathematical Reviews MR0277490 (43 #3223).

This book may be regarded as a sequel to the author's Introduction to analytic number theory (1968). The arithmetical functions considered in the present book are those associated with the distribution of prime numbers as well as the partition function and the divisor function. The book is written in a very lucid style but the statement in the preface that "in order to read the book only a modicum of acquaintance with analysis and number theory is required" is not devoid of optimism.

There are eight chapters. (I) The prime number theorem and Selberg's method. (II) The zeta-function of Riemann. (III) Littlewood's theorem and Weyl's method. (IV) Vinogradov's method. (V) Theorems of Hoheisel and of Ingham. (VI) Dirichlet's $L$-functions and Siegel's theorem. (VII) Theorems of Hardy-Ramanujan and of Rademacher on the partition function. (VIII) Dirichlet's divisor problem.

Each chapter ends with extensive notes mentioning the authors and the sources of the theorems dealt with, referring to related formulas, other proofs, etc., and containing a large amount of detailed information, making the book very lively and interesting. Many references go back to 19th century results and methods, but most of them relate to recent work, even to a paper published in 1968 [the author and R Narasimhan (1968)].

The content of this volume is more extensive than that of the author's introductory book mentioned above which already excelled in the richness of its material. The author's way of presenting the material in these two volumes makes it feasible for a reader to get acquainted with the masterpieces of analytical number theory without having to cover too much ground.

5.3. Review by: C Benedetti.
Genus 27 (1/4) (1971), 426-428.

One of the oldest and most fascinating branches of mathematics, Number Theory, or Higher Arithmetic, or simply Arithmetic, has for about a century become one of the most abstruse and almost incomprehensible to the non-specialist. One could not help but be dazzled and enchanted by the propositions established more than two millennia ago by Euclid, Eratosthenes, and other classics on the subject, and by their clear and natural proofs. With the most rudimentary techniques based on the four elementary operations, these ancient authors managed to establish key propositions in arithmetic. Suffice it to recall Euclid's famous proof that the number of primes is infinite, his famous algorithm for determining the greatest common divisor, the Sieve of Eratosthenes, etc.

Such proofs remain a model of simplicity and power. Over time, results in arithmetic have become increasingly difficult to obtain, and at the same time, increasingly refined methods and techniques have become widespread.

We do not know, however, to what extent successes became rarer due to the exhaustion of the most readily available fields or because the simple and penetrating visions of the greats of antiquity were abandoned in favour of increasingly taking advantage of the more sophisticated mathematical techniques that were gradually entering the work of new generations of mathematicians.

However, it can be assumed that after the first classical and fundamental results of the type mentioned, obtained with extremely simple, if not naive, means, the subsequent results, often less notable (with the exception of the contributions of Gauss, Euler, Lagrange and some others) are almost always obtained at the cost of ever-increasing and sometimes even disconcerting efforts (just consider the long and tormented demonstrations of I M Vinogradov in additive arithmetic). Furthermore, many of these laborious proofs based on intricate notions of the theory of functions of complex variables are aimed at establishing arithmetic properties that hold on average or are asymptotic in nature, that is, despite these enormous efforts we must be content with results that are incomplete and imprecise, or attenuated by various disclaimers regarding their general validity.

Gauss is said to have said, "Mathematics is the queen of science and arithmetic is the queen of mathematics," and this attraction of arithmetic has always been felt by the most original and combative mathematicians. Unfortunately, many simple and plausible arithmetical propositions, immediately verifiable experimentally, often resist every attempt at proof, and then a whole host of mathematicians rages around these propositions, unleashing the most powerful and modern weapons. Thus, around simple arithmetic problems, a mass of specialised literature grows, becoming increasingly complicated and in which only a small group of highly skilled specialists can find their way. This often continues until some young mathematician, perhaps an amateur, points to a new, simpler and more original approach that allows these problems to be solved or to come significantly closer to their solutions.
...
The book we are about to discuss can be considered the sequel to another volume by the same author entitled: Introduction to Analytical Number Theory, published by the same publisher. The immediately evident merit of these two books is the clarity of the exposition facilitated by the adoption of a non-tiring, classical symbology. In the volume we are dealing with, the results connected to the study of the aforementioned Riemann function $\zeta(x)$ are mostly presented. In fact, after having reported Selberg's proof of the so-called "prime number theorem", which, unlike those of Hadamard and De La Vallée Poussin, does not include functions of a complex variable, we proceed it can be said that a large part of the book deals directly or through collateral questions with the study of the "zeros" of $\zeta(x)$ in the famous "critical range $0 ≤ R(x) ≤ 1$.

Various results are reported. In addition to Riemann's classic formulas, the now classic contributions of von Mangolt, Landau, Hardy, Littlewood, Weyl, Vinogradov, and Chudakov are presented. The final chapters deal with Dirichlet series, the results of Hardy, Ramanujan, and Rademacher, and the problem of Dirichlet divisors with the results of Perron, Voronoi, Hardy, and Ingham.

While the book, as we've already mentioned, is appealing even to non-specialists due to its symbolism and clear presentation, it nevertheless deals with rather difficult topics requiring a good basic understanding of the theory of functions of complex variables and familiarity with an elementary textbook on number theory, such as the well-known Hardy and Wright textbook. Many of the results reported in the book are scattered throughout more difficult monographs or original articles. Appropriate notes at the end of each chapter help the reader locate elements assumed to be familiar during the various proofs, while also providing helpful bibliographical references for orienting oneself in the literature related to the topics covered in the volume.

If one were to criticise this work, it would be a certain, though difficult to eliminate, fragmentary nature with which the various arguments are presented. The non-specialist in number theory (which the author of this book, I believe, did not wish to exclude from his readership) needs a certain guiding thread that allows him to glimpse the principal goals toward which many of these results and methods are aimed. For example, the exposition of some of the most significant objectives that could be achieved, as a more or less immediate consequence, through the proof of the famous Riemann hypothesis would contribute to creating a guiding thread, to standardising all those efforts relating to the search for the zeros of $\zeta(x)$ in the critical range, channelling all the results toward points of arrival whose importance is more easily recognised. But aside from these considerations about the appropriateness of finalising the methods presented, for reasons of stimulation and interest for neophytes and the general non-specialist reader, the book seems to us to reconcile the requirements of clarity with the necessary rigour, and this is truly not easy in subjects such as these. Springer's typography is, as usual, superb.

5.4. Review by: H M Stark.
Bulletin of the American Mathematical Society 77 (6) (1971), 943-957.

This brings us to the same author's introduction to analytic number theory, which he has entitled Arithmetic functions. About two-thirds of the book is devoted to the distribution of primes and the rest is split between the partition function and the divisor function. Chapter One presents an elementary proof of the prime number theorem. The proof chosen follows that of Wirsing, which is capable of producing an error term but only the asymptotic result is given here.
...
In my opinion, the main value of the book lies in Chapters 1-5 on the distribution of primes. These chapters certainly provide an updating of Ingham's Cambridge Tract, The distribution of prime numbers (although much less so when it is combined with Titchmarsh's book, The theory of the Riemann zeta function). Without the corresponding work on primes in arithmetic progressions, Siegel's theorem seems isolated and, in particular, it is difficult to appreciate the ineffective nature of the result. Sieve results are not presented even though they are at present producing most of the new results in the subject - e.g. Montgomery's improvement of Ingham's theorem "by a new method". There are notes given in each chapter to further references and original sources which should prove useful.

6. Elliptic Functions (1985), by K Chandrasekharan.

6.1. From the Preface.

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

Professor Raghavan Narasimhan has read the definitive English version of the text, and made illuminating comments, as a result of which I have improved the presentation in several places. Dr Anton Good has looked through the first German version, and spared the time for many useful discussions. Dr Peter Thurnheer, who had attended the course, helped me check the detailed calculations that lurk behind some of the statements in the text. Mr Albert Stadler has assisted me in tracing the bibliographical references and in proof-reading. My sincere thanks go to them all for sustaining this effort during a difficult twelvemonth.

Thanks are due to the editors of the Grundlehren, particularly to Professor Beno Eckmann, and to Springer-Verlag, for getting this thing into print.

6.2. Review by: Marvin I Knopp.
Mathematical Reviews MR0808396 (87e:11058).

Taking for granted "the fundamentals of complex analysis" and starting with a brief discussion of meromorphic periodic functions, the author in this book develops a full-blown introduction to the deeper properties of elliptic functions, theta-functions and modular forms. Or, more properly, he allows it to evolve, for his treatment has a natural flow and an organic unity that impart to it the feeling of having written itself. This raises the possibility, of course, that a reader not previously familiar with the subject might confuse the clarity of exposition for simplicity (or even shallowness) of content. However, the intrinsic power of this material combines with the author's sense of mathematical history and grasp of recent developments - evident in the detailed "Notes" appended to each chapter - to negate this danger. Thus, notwithstanding the recent spate of books (some of them excellent) dealing with modular functions, Elliptic functions could well be the best choice for many - professional mathematicians and graduate students - seeking an introduction to the subject. This is particularly true since the work contains far more about elliptic functions than do most other books on modular functions and vice versa.
...
In the breadth, depth and inevitability of treatment of this beautiful material, the author has made a contribution to the mathematical community consistent with the distinction of his career. That he has succeeded in compressing this treatment into a succinct monograph of fewer than 190 pages is a testament to his taste, discipline and powers of exposition.

7.1. From the Preface.

In grateful remembrance of Marston Morse and John von Neumann.

This text formed the basis of an optional course of lectures I gave in German at the Swiss Federal Institute of Technology (ETH), Zürich, during the Wintersemester of 1986-87, to undergraduates whose interests were rather mixed, and who were supposed, in general, to be acquainted with only the rudiments of real and complex analysis. The choice of material and the treatment were linked to that supposition. The idea of publishing this originated with Dr Joachim Heinze of Springer-Verlag. I have, in response, checked the text once more, and added some notes and references. My warm thanks go to Professor Raghavan Narasimhan and to Dr Albert Stadler, for their helpful and careful scrutiny of the manuscript, which resulted in the removal of some obscurities, and to Springer-Verlag for their courtesy and cooperation. I have to thank Dr Stadler also for his assistance with the diagrams and with the proof-reading.

7.2. Review by: Colin C Graham.
Mathematical Reviews MR0978387 (90c:42001).

The author gives a clear and concise presentation of classical results on one-variable Fourier transforms. There are three chapters. Chapters I and II assume Lebesgue integration. Chapter III assumes Riemann-Stieltjes integration. Chapter I covers the $L^{1}$ theory and includes Mellin transforms, the central limit theorem, Tauberian theory, applications to the heat and Laplace equations, and a few pages on transforms in several variables (emphasis on radial functions).

Chapter II covers the $L^{2}$ theory and includes material on Heisenberg's inequality, Hardy's characterisation of functions which together with their Fourier transform are $O(e^{−x^{2}})$, the Paley-Wiener theorem, Bernstein's theorems on the size of the derivative of a transform and Fourier series, and an outline of the proof of the Plancherel theorem in several variables.

Chapter III covers Fourier-Stieltjes transforms, using the point of view of derivatives of nondecreasing functions and includes a characterisation of measurable positive-definite functions (with a separate proof for the continuous case), and a uniqueness theorem for transforms of locally integrable functions (due to A C Offord (1936)).

There are no exercises.

7.3. Review by: D H Griffel.
The Mathematical Gazette 74 (467) (1990), 88.

In his recent book Fourier analysis, T W Körner writes:

Oxford stories lose little in the telling. Titchmarsh wrote of Hardy that: "I worked on the theory of Fourier integrals under his guidance for a good many years before I discovered that this theory has applications in applied mathematics, if the solution of certain differential equations can be called 'applied'." Changes in the undergraduate curriculum have ensured that few pure mathematicians today can enjoy such a sheltered upbringing. ...

Perhaps things have not changed as much as all that. Professor Chandrasekharan's book is based on lectures to "undergraduates whose interests were rather mixed, and who were acquainted with only the rudiments of real and complex analysis"; yet it is written very much in the Hardy spirit, being an elegant and concise exposition of the theory of Fourier integrals regarded strictly as a branch of classical analysis. There are a few applications. For example, there is a proof of Hardy's result that (under certain assumptions) the value of a function $f$ at a point $z$ is determined in terms of its values at the integers by

         $f(z) = \sum^{n=\infin}_{n=-\infin} f(n) \large \frac {sin \pi (z-n)}{\pi (z-n)}$.

There are proofs of the "central limit theorem" and of "Heisenberg's inequality", with no indication of their statistical and quantum-mechanical meaning. And there is a proof of the Cauchy-Poisson formula for the solution of the wave equation, but no hint of the general utility of Fourier methods in linear partial differential equations.

What you do get for your money is a fine exposition of the theory of Fourier integrals in $L_{1}$ and $L_{2}$, and a Stieltjes-integral version. There are historical notes, and references to recent results and open questions. The "rudiments of analysis" assumed known by the reader include a thorough knowledge of Riemann-Stieltjes and Lebesgue integration. Page 1 introduces the $L_{p}$spaces, regarded as Banach spaces of equivalence classes of functions, with a remark that these spaces will not usually be distinguished from the vector spaces whose elements are functions. This sets the tone for the rest of the book; the exposition is always precise and clear, and proofs are given in full, but there are no unnecessary words, and unsophisticated readers will find it tough going.

The treatment is quite strictly classical: no Fourier analysis on groups, and no generalised functions (though one sometimes senses generalised transforms lurking behind the scenes, influencing the theory, rather as poles off the axis influence the values of an analytic function on the real axis). If you know about Fourier transforms and want to learn the analytical details, this is the book for you. But if you are looking for an undergraduate textbook which motivates and inspires, then ignore the publisher's designation of this volume as a "universitext", and look elsewhere.

7.4. Review by Allen Stenger.
Mathematical Association of America Reviews (12 December 2017).

This is a concise and very focused introduction to the theory of Fourier transforms of functions defined on the real line. The prerequisites are a modest grasp of Lebesgue integration (mostly the convergence theorems); it's also helpful to be familiar with Fourier series, as the book uses these for motivation in some cases.

This is primarily a pure-math view of the subject. It does show how Fourier transforms can be used to show the existence and uniqueness of solutions to two heat equations (differential equations), but does not show how to solve them. It also gives applications to other pure-math areas, such as the central limit theorem and Tauberian theorems.

The first half of the book (Chapter 1) deals with the theory for $L^{1}$ functions. The theory is most straightforward in this case, and many of the uses of Fourier transforms are for $L^{1}$ functions. The main drawback is that the transform of an $L^{1}$ function may not be $L^{1}$, so the transform is not always invertible. Most of the rest of the book (Chapter 2) deals with $L^{2}$ functions; the theory is more symmetric here because the transforms are also $L^{2}$ functions, but it is harder to get started because the naive approach of defining the transform as an integral doesn't always converge for$L^{2}$ functions, and the definition has to be bootstrapped. This chapter depends heavily on the $L^{1}$ theory. The rest of the book (Chapter 3) looks briefly at Fourier-Stieltjes transforms, a generalisation of the $L^{1}$ theory.

I like this book because it is modern without being abstract. The "classical" in the title means both that it sticks to the real line, and that it (generally) avoids the Banach algebra approach. Most treatments of Fourier transforms are aimed at the applications in physics, and do not go into as much depth into the theory as this book does. Another good but extremely concise treatment (that does include Banach algebras) is in Rudin's Real and Complex Analysis. An older book that goes into even more depth is Titchmarsh's Introduction to The Theory of Fourier Integrals. Wiener's The Fourier Integral and Certain of its Applications has a lot of overlap with this book, but is not as balanced and focuses on Wiener's research interests.

8. A Course on Topological Groups (1996), by K Chandrasekharan.

8.1. From the Author's Note.

This course has for its aim a proof of the Peter-Weyl theorem (1927), that every complex-valued continuous function on a compact topological group is a uniform limit of finite linear combinations of representation functions coming from irreducible representations. The method of proof adopted here is the one expounded by Warren Ambrose in his MIT lectures (1952). It incorporates the ideas originally introduced in this context by John von Neumann, and André Weil, and makes use of the $L_{2}$-algebra of the group relative to Haar measure. The topological, analytical, and algebraic groundwork needed for the proof is provided as part of the course.
...
It is Ambrose's approach that is the prime influence in this presentation, which was offered as an optional course at the ETH, Zürich, more than once, during the years 1965-88.

8.2. Review by: Joseph Max Tosenblatt.
Mathematical Reviews MR1661135 (99k:22010).

This short book introduces the concept of locally compact groups and constructs the Haar measure in the first two chapters. The spectral theorem for compact operators is proved in the third chapter. Representations of compact groups is the topic of the fourth chapter. The presentation in this text is clear and to the point. The methods used are good classical ones.

This is a good text for a student who knows little about locally compact groups and wants to get an introduction to some of the fundamental ideas needed to begin the study of them.

8.3. Review by: Michael Berg.
Mathematical Association of America Reviews (5 April 2012).

On p. 89 of the book under review we find the following:

Our aim is to prove the following results on finite-dimensional representations of compact groups ... (1) that every representation is equivalent to a unitary representation, (2) that every representation is completely reducible, (3) the orthogonality relations, and (4) the Peter-Weyl theorem.

With these remarks Chandrasekharan introduces the fourth and final chapter of this compact book on topological groups, indeed on the according representation theory. With the author, now 91, a mainstay at both the Tata Institute and the ETH in Zürich as well as a major figure in number theory, the book enjoys the advantage of a certain terseness, great accessibility, and an implicit focus on the applications of this material to other parts of mathematics.

Clearly Chandrasekharan is keen on getting somewhere fast, and he is willing to dispense with aspects of the theory that don't serve his purpose. In fact, the book starts off with an "Author's Note" to the effect that his "aim is a proof of the Peter-Weyl theorem (1927) that every complex-valued continuous function on a compact topological group is a uniform limit of finite linear combinations coming of representation functions coming from irreducible representations." With that Chandrasekharan appends a list of sources (all classics) and singles out von Neumann and Weil as the originators of the circle of ideas used in the upcoming treatment which, in turn, is based on Ambrose's 1952 MIT lectures.

And then he is off to the races: the first three chapters deal with, respectively, the obligatory topological preliminaries, a development of the theory of Haar measure on an LCA group, and a discussion of Hilbert spaces and the spectral theorem. Regarding the latter, Chandrasekharan presents a particularly accessible statement in five parts that tells the story very well and leaves little room for confusion: $T$ is a real, completely continuous operator on a Hilbert space with eigenvalues $\lambda n$; then

- for all positive , there are only finitely many $\lambda n$ with $|\lambda n| > \epsilon$, whence (easy exercise) we get only a countable list of $\lambda n$, and $\lambda n \rightarrow 0$;

- each nonzero eigenvalue has a finite dimensional eigenspace;

- the full Hilbert space is the direct sum of these eigenspaces;

- the closure of $T$'s image is the sum of the eigenspaces coming from the non-zero eigenvalues; and

- an operator $S$ on the same Hilbert space commutes with $T$ iff $S$ leaves each of $T$'s eigenspaces invariant.

This explicit enumeration of properties of the eigenvalues of $T$ is obviously very useful and makes for great clarity.

In point of fact, great clarity is one of the virtues particularly in evidence as far as this book is concerned: it is a pleasure to read, and qualifies as a particularly good source for learning this important material - or to use as a text in a course.

Finally, just for good measure, here is the very last sentence of the book (cf. p. 114): "Every locally Euclidean group is isomorphic (group isomorphism and homeomorphism of the space) to a Lie group (Hilbert's Fifth Problem, 1900). [Proved by D Montgomery, L Zippin, and A Gleason, 1952–1953.] It does not get much prettier than that.

9.1. From the Publisher.

This book on integration theory is based on the lecture notes for courses that the author gave at the Tata Institute of Fundamental Research, Mumbai, and at ETH, Zurich. The subject matter is classical. The goal of the notes is to provide a concise, clear, and accurate treatment of the basic ideas of the subject.

9.2. From the Author's Note.

The material of this course is classical. The approach adopted dates back to Daniell, and was used, for instance, by Irving E Segal in his Chicago lectures (1950). Differences in the treatment of a subject like this lurk largely in the details. The goal here is concision, clarity and accuracy.
...
The influence, particularly of W Ambrose, Lectures on topological groups, P R Halmos, Measure Theory, and I E Segal, Introduction to modern integration theory on this presentation must be obvious.

9.3. Review by: Robert Gardner Bartle.
Mathematical Reviews MR1662921 (99k:28001).

As the author asserts, the material presented in this slim volume is classical; his goal has been "concision, clarity, and accuracy".

There are five chapters; the first (and longest one) is titled "Integration on a measure space". A simple function is a finite linear combination of characteristic functions of sets with finite measure. An integrable function is the point-wise limit of a sequence of simple functions whose integrals are Cauchy in mean. After establishing that the integral is well-defined, the author proves the monotone convergence and Lebesgue dominated convergence theorems.

Chapter 2 deals with the Lebesgue spaces $L_{p}, 1 ≤ p ≤ ∞$. The relevant inequalities and the completeness of $L_{1}$ are proved. In the next chapter, the author discusses outer measures, and establishes the Hahn-Kolmogorov extension theorem and the existence of Lebesgue measure on $\mathbb{R}$. In Chapter 4 he treats product measures and Fubini's theorem. In the final chapter, he proves the theorems of Hahn, Jordan, Lebesgue and Radon-Nikodým. The connection is made between certain measures on R and functions of bounded variation and absolute continuity.

The author is extraordinarily careful in detail, for example, in showing that the integrals of simple functions and integrable functions are well-defined. Unfortunately, there are no exercises or historical comments.

9.4. Review by Allen Stenger.
Mathematical Association of America Reviews (14 April 2012).

This is a conventional development of integration theory, starting with measure spaces. The Author's Note (p. vii) states that "The goal here is concision, clarity, and accuracy," and the book meets these goals. By American standards this is not a "Course", as it has no exercises or motivation, and only a modest set of examples. The present volume is a reprint of the original 1996 publication.

I think this book works best as a review or as an introduction to the more abstract theories of integration. It is too abstract for a first course in integration, as it starts immediately with the most abstract theories and only (a great deal later) deals with integration on the real line. A book that takes a similar approach, but with more exercises and more concreteness, is Hewitt & Stromberg's Real and Abstract Analysis.