# Peter Borwein's books

We list below books written or edited by Peter Borwein. We give extracts from different sources: the publisher's description, the preface to the book, and reviews of the book.

Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987) with Jonathan M Borwein

A Dictionary of Real Numbers (1990) with Jonathan M Borwein

Polynomials and polynomial inequalities (1995) with Tamas Erdelyi

Pi: A Source Book (1997) with Lennart Berggren, Jonathan M Borwein

Computational Excursions in Analysis and Number Theory (2002)

The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike (2007) with Stephen Choi, Brendan Rooney and Andrea Weirathmueller

Selected Writings on Experimental and Computational Mathematics (2010) with Jonathan M Borwein

Mathematicians on Creativity, (editor) with Peter Liljedahl and Helen Zhai

**Click on a link below to go to that book**Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987) with Jonathan M Borwein

A Dictionary of Real Numbers (1990) with Jonathan M Borwein

Polynomials and polynomial inequalities (1995) with Tamas Erdelyi

Pi: A Source Book (1997) with Lennart Berggren, Jonathan M Borwein

Computational Excursions in Analysis and Number Theory (2002)

The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike (2007) with Stephen Choi, Brendan Rooney and Andrea Weirathmueller

Selected Writings on Experimental and Computational Mathematics (2010) with Jonathan M Borwein

Mathematicians on Creativity, (editor) with Peter Liljedahl and Helen Zhai

**1. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987), by Jonathan M Borwein and Peter B Borwein.**

**1.1. Note.**

Reprinted 1988, 1996, Chinese edition 1995, paperback 1998.

**1.2. On the Back of the Book.**

Critical Acclaim for Pi and the AGM:

"Fortunately we have the Borweins' beautiful book ... explores in the first five chapters the glorious world so dear to Ramanujan ... would be a marvellous text book for a graduate course." - Bulletin of the American Mathematical Society.

"What am I to say about this quilt of a book? One is reminded of Debussy who, on being asked by his harmony teacher to explain what rules he was following as he improvised at the piano, replied, 'Mon plaisir.' The authors are cultured mathematicians. They have selected what has amused and intrigued them in the hope that it will do the same for us. Frankly, I cannot think of a more provocative and generous recipe for writing a book ... (it) is cleanly, even beautifully written, and attractively printed and composed. The book is unique. I cannot think of any other book in print which contains more than a smidgen of the material these authors have included." - SIAM Review.

"If this subject begins to sound more interesting than it did in the last newspaper article on 130 million digits of

*π*, I have partly succeeded. To succeed completely I will have gotten you interested enough to read the delightful and important book by the Borweins." - American Mathematical Monthly.

"The authors are to be commended for their careful presentation of much of the content of Ramanujan's famous paper, 'Modular Equations and Approximations to Pi'. This material has not heretofore appeared in book form. However, more importantly, Ramanujan provided no proofs for many of the claims that he made, and so the authors provided many of the missing details ... The Borweins, indeed have helped us find the right roads." - Mathematics of Computation.

**1.3. From the Preface.**

A central thread of this book is the arithmetic-geometric mean iteration of Gauss, Lagrange and Legendre. A second thread is the calculation of

*π*. The two threads are intimately interwoven and provide a remarkable example of the application to twentieth-century computational concerns of the type of nineteenth-century analysis whose neglect Klein so deplores. The calculation of digits of

*π*has had a fascination that has far exceeded utilitarian concerns - a fascination that has driven some to dedicate their lives to calculations we may now electronically effect in seconds. The methods that make the computation of hundreds of millions of digits of or any elementary function within our grasp are rooted in the AGM and this is where our interest in the material began. Making sense of this material took us in three directions and motivated our writing this book.

The first direction leads to nineteenth-century analysis and in particular the transformation theory of elliptic integrals. This necessitates at least a brief discussion of a number of topics including elliptic integrals and functions, theta functions, and modular functions. These attractive and once central concerns of analysis have been dropped from the standard curriculum - and much that is beautiful has become relatively inaccessible except to the expert or the archivist. In presenting this material we have not striven for generality. This is available in the specialty literature. At times we have settled for giving only a taste of the material and a few pointers on where it can be pursued.

We have found this excuse to consult the nineteenth-century masters a pleasurable and rewarding bonus - as Hermann points out in his introduction to Klein, "We are so used to thinking in terms of the 'progress' of science that it is hard for us to remember that certain matters were better understood one hundred years ago."

The second direction takes us into the domain of analytic complexity. How intrinsically difficult is it to calculate algebraic functions, elementary functions and constants, and the familiar functions of mathematical physics? Here part of the attraction is the surprising answers - the familiar methods are often far from optimal.

Finally, an honest treatment invites exploration of applications and ancillary material, particularly the rich and beautiful interconnections between the function theory and the number theory. Included, for example, are the Rogers-Ramanujan identities; algebraic series for

*π*; results on sums of two and four squares; the transcendence of

*π*and

*e*and a discussion of Madelung's constant, lattice sums, and elliptic invariants.

Our primary concern throughout has been the interplay of analysis and mathematical application. We hope we have elucidated a variety of useful and attractive analytic techniques. This book should be accessible to any graduate student. Only rarely does it assume more than the content of undergraduate courses in real and complex analysis. It is, however, at times terse, at times computational, and some of the exercises are difficult. A fair amount of the material, particularly on the approximation and computation of

*π*and the elementary functions, is new and only partially available in research papers.

**1.4. Review by: H London.**

*Mathematical Reviews*MR0877728

**(89a:11134)**.

This book reveals the close relationship between the algebraic-geometric mean iteration and the calculation of

*π*.

The topic of algebraic-geometric mean iteration leads to a discussion on the theory of elliptic integrals and functions, theta functions and modular functions. The calculation of

*π*leads into the area of calculating algebraic functions, elementary functions and constants plus the transcendence of

*π*and

*e*. The calculation of

*π*advanced with a frenzy with the advent of modern computers. Briefly, in 1706 the first 100 digits of

*π*were calculated. By 1844 the first 205 digits were known. In 1947 the first 808 digits of π were computed using a desk calculator.

Then the modern computer came on the scene. Now the known first digits changed rapidly ...

**1.5. Review by: Richard Askey.**

*Amer. Math. Monthly*

**95**(9) (1988), 895-897.

The Mathematical Association of America, The American Mathematical Society and the Society for Industrial and Applied Mathematics have joined to try to get better media coverage of mathematics. One result of this is stories that say that computers have calculated 10 million, or now 130 million digits of

*π*. When asked to comment on this by friends, most mathematicians wince and say they have no idea why anyone would want to do these calculations. Privately, they start to have serious doubts about more publicity for mathematics. If the computation of

*π*to millions of places is to be the image of a mathematician, then it might be better to keep mathematics out of the newspaper. However, ten million is so much larger then the 707 digits that was claimed to be known when I first heard of this problem that something interesting must lie behind these calculations. Part of the improvement comes from larger and faster machines, but much more comes from increased mathematical knowledge. Part of this is specific to certain numbers, and part is general and of wide applicability. All of it is very interesting.

...

... the next time you read a newspaper story about mathematics, read it the way political news needs to be read in many parts of the world, that is, read between the lines to get the story that is really there. It is probably very interesting, unlike the story that appeared in the paper. The problem of educating a few reporters so that they appreciate and understand some mathematics can probably be solved, but the problem of educating the editors to realise that not only does mathematics matter, but they should treat it seriously is much harder. However, an existence theorem exists at the Los Angeles Times, both for a reporter and an editor.

**1.6. Review by: Bruce C Berndt.**

*Mathematics of Computation*

**50**181 (1988), 352-354.

When the reviewer was a teenager, he and three friends, after a high school basketball game, would frequently get in a car and drive over the labyrinth of country roads in the rural area in which they lived. The game we played was to guess the name of the first village we would enter. Since there were many meandering roads and countless small hamlets that dotted the rural landscape, since it was dark, and since we were not blessed with keen senses of direction, we were often surprised when the signpost identified for us the town that we were entering.

For one not too familiar with the seemingly disparate topics examined by the Borweins in their book, one might surmise that the authors were travelling along mathematical byways with the same naiveté and lack of direction as the reviewer and his friends. However, the authors travel along well-lit roads that are marked by the road signs of elegance and usefulness and that lead to beautiful results. They do not take gravel-surfaced roads that lead to dead ends in cow pastures. But sometimes the destinations are surprising - at least to those not familiar with the landscape.

**1.7. Review by: Peter Cass.**

*The Mathematical Gazette*

**83**(497) (1999), 334-335.

This exciting and well-written book is the product of the sound scholarship and enormous enthusiasm of its authors. The subtitle, 'A study in analytic number theory and computational complexity', captures its central theme very well.

...

In exploring their principal themes, the arithmetic-geometric mean (AGM) iteration of Gauss, Lagrange, and Legendre, and the calculation of

*π*, the authors have gathered together a wealth of largely forgotten treasures from nineteenth-century analysis, drawn from the works of many masters, notably Gauss, Jacobi and Weierstrass. They have set them in a modern context, and directly related them to current research. For example, the transformation theory of elliptic functions, Jacobi's triple product, theta functions, and modular equations and algebraic approximations are discussed extensively. Chapter 3 contains welcome material elucidating some of Ramanujan's work, in particular the Rogers-Ramanujan identities.

...

I believe that most mathematicians will find much of interest in this engaging and literate book - a work of real enduring value.

**1.8. Review by: Jet Wimp.**

*SIAM Review*

**30**(3) (1988), 530-533.

What am I to say about this quilt of a book? One is reminded of Debussy who, on being asked by his harmony teacher to explain what rules he was following as he improvised at the piano, replied, "Mon plaisir." The authors are cultured mathematicians. They have selected what has amused and intrigued them in the hope that it will do the same for us. Frankly, I cannot think of a more provocative and generous recipe for writing a book. Intellectually, what can we really offer one another more genuine than our own excitement? - for, as much as some pretend otherwise, there is no trustworthy yardstick for the value of a piece of mathematics.

The book is cleanly, even beautifully, written, and attractively printed and composed. The book is unique. I cannot think of any other book in print which contains more than a smidgen of the material these authors have included. It is self-contained and copiously indexed. Each section concludes with exercises which invite the reader to share in the experience. These impressive exercises - really, excursions - are abundant, challenging, and selected so as to impinge on many subjects not discussed in the text. Can the book be used as a text? I recoil at such questions. But, yes. The book is a text if, for a little while, you are willing to be a student. And I suspect that if you purchase it you will not soon put it down.

**1.9. Review by: George E Andrews.**

*Bull. Amer. Math. Soc.*

**22**(1) (1990), 198-201.

P Beckmann wrote in April 1987 on the work of Y Kanada et al. calculating the first 140 million digits of

*π*:

It takes a lot of brains to program a computer to calculate (and verify!) 140 million decimal digits in a reasonable time, and the programmers have my respect and admiration - but they have it as sportsmen, not as scientists. The need for brains, or even the development of new techniques, is not what defines science. It also takes brains and the development of new techniques to stuff 21 people into a phone booth. And indeed, the calculation of the first 140 million digits is a feat worthy of ranking along with the other stunts in the Guinness Book of World Records. But I believe that when one applies the test of whether it can lead to a generalisation of knowledge, this feat will be found scientifically worthless.Surely if Beckmann's assertions are correct, we should be disturbed by the frivolous nature of such work ... Fortunately we have the Borweins' beautiful book to refute Beckmann's argument. There is both serious and beautiful mathematics underlying these massive calculations of the digits of

*π*.

...

It seems to me that this would be a marvellous text book for a graduate course in what has sometimes been called constructive mathematics, that never-never land between mathematics and computer science. This book contains important and currently neglected topics in classical analysis together with enough material on algorithms to make it a really exciting new course. The authors surely had something like this in mind for they provide ample exercises. I strongly hope that a number of people will be inspired to use the Borweins' book as the text for such a course.

**1.10. Review by: Allen Stenger.**

*Mathematical Association of America*(16 July 2018).

This is a very erudite book, and a large part of its charm is that it shows us many unexpected connections between different parts of mathematics. The AGM of title is the Arithmetic-Geometric Mean, a recurrence that was initially studied by Lagrange and Gauss in connection with numerical approximations to elliptic integrals. The book is inspired to some extent by Ramanujan's 1914 paper,

*Modular Equations and Approximations to Pi*. Like much of Ramanujan's work, this paper is full of interesting ideas but skimpy on proofs, and the present book is the first time that these ideas were worked out in detail and proved.

The publisher classifies this book as number theory, and there is some truth to that. It also deals with many of the topics of analysis from the late 1800s and the early 1900s, in particular theta functions and modular groups, and the last half of the book is concerned mostly with numerical approximations for various transcendental functions, and calculations for

*π*are spread throughout the book.

This summary makes the book sound like a hodgepodge, but in fact it hangs together very well. Roughly the first half of the book deals with elliptic integrals, theta functions, and the AGM. This part includes materials on partitions, the Rogers–Ramanujan identities, and representing numbers as sums of squares. It also includes the formulas and algorithms for calculating

*π*. The second half of the book is focused more on computer arithmetic and complexity of calculation, including results on fast Fourier transforms and fast multiplication. It also includes the history of

*π*calculations and results on Diophantine approximations (irrationality measures) of

*π*. About $\large\frac{1}{4}\normalsize$ to $\large\frac{1}{3}\normalsize$ of the book is exercises, so it would work well as a textbook or problem book as well as a monograph (though there are no solutions; some exercises have hints).

...

Bottom line: the book is a tour de force. If you are interested in any of topics covered, it will lead you along somewhat-winding paths to other topics you'll like. Even if you're not interested, take a look and marvel at how closely connected its many diverse topics are.

**2. A Dictionary of Real Numbers (1990)**,

**by Jonathan M Borwein and Peter B Borwein.**

**2.1. Note.**

Published by different publishers in each of 1991, 1992, 1993 and 2011.

**2.2. From the Preface.**

How do we recognise that the number .93371663 ... is actually $2 \log_{10} \large\frac{1}{2}\normalsize (e + \pi)$? Gauss observed that the number 1.85407467 ... is (essentially) a rational value of an elliptic integral - an observation that was critical in the development of nineteenth century analysis. How do we decide that such a number is actually a special value of a familiar function without the tools Gauss had at his disposal, which were, presumably, phenomenal insight and a prodigious memory? Part of the answer, we hope, lies in this volume.

This book is structured like a reverse telephone book, or more accurately, like a reverse handbook of special function values. It is a list of just over 100,000 eight-digit real numbers in the interval [0,1) that arise as the first eight digits of special values of familiar functions. It is designed for people, like ourselves, who encounter various numbers computationally and want to know if these numbers have some simple form. This is not a particularly well-defined endeavour - every eight-digit number is rational and this is not interesting. However, the chances of an eight digit number agreeing with a small rational, say with numerator and denominator less than twenty-five, is small. Thus the list is comprised primarily of special function evaluations at various algebraic and simple transcendental values. The exact numbers included are described below.

Each entry consists of the first eight digits after the decimal point of the number in question. The values are truncated not rounded. The next part of the entry specifies the function, and the final part of the entry is the value at which the function is evaluated. So -4.828313737... is entered as "8283 1373 ln : $5^{-3}$". The abbreviations are also described below. There are two exceptions to this format. One is to describe certain combinations of two functions, for instance "8064 9591 exp(√3) + exp(

*e*)" the meaning of which is self-evident. The other is for real roots of cubic polynomials, i.e. "2027 1481 rt: (5, 8, -8, -3)" means that the polynomial $5x^{3} + 8x^{2} - 8x - 3$ has a real root, the fractional part of which is .20271481 .... Repeats have in general been excluded except for cases where two genuinely different numbers agree through at least eight digits. These latter coincidences account for the fewer than fifty repeat entries.

If the number you are checking is not in the list try checking one divided by the number and perhaps a few other variants such as one minus the number. Of course, finding the number in the list, in general, only indicates that this is a good candidate for the number. Proving agreement or checking to further accuracy is then appropriate.

**2.3. Review by: Nick Lord.**

*The Mathematical Gazette*

**74**(470) (1990), 395-396.

This large book is precisely what its title suggests-an impressive list of some 100,000 numbers between 0 and 1 (in order, to 8 decimal places) which are of the form $x - [x]$ for interesting or significant real numbers $x$. By "interesting or significant" is meant either evaluations of standard functions at special points or simple combinations of significant numbers (including a sprinkling of physical constants).

...

The authors suggest that such a collection of numbers might be useful in research if you encounter a number whose value you know accurately but which you suspect to have deeper significance; agreement to 8 decimal places with an entry in the table reduces the risk of misappropriation and may reveal an unexpected connection. And, indeed, I was able to recapture something of the excitement caused by Euler's evaluation of $\Sigma \large\frac{1}{n^{2}\normalsize}$ by turning up (1) .6449 3406 = $\large\frac{\pi^2}{6}$. However, except perhaps as an example of an enterprise which could not seriously have been contemplated before the advent of computers, I cannot really see a place for this book other than in a university library.

**3. Polynomials and polynomial inequalities (1995), by Peter Borwein and Tamas Erdelyi.**

**3.1. From the Publisher.**

After an introduction to the geometry of polynomials and a discussion of refinements of the Fundamental Theorem of Algebra, the book turns to a consideration of various special polynomials. Chebyshev and Descartes systems are then introduced, and Müntz systems and rational systems are examined in detail. Subsequent chapters discuss denseness questions and the inequalities satisfied by polynomials and rational functions. Appendices on algorithms and computational concerns, on the interpolation theorem, and on orthogonality and irrationality round off the text. The book is self-contained and assumes at most a senior-undergraduate familiarity with real and complex analysis.

**3.2. From the Preface.**

Polynomials pervade mathematics, and much that is beautiful in mathematics is related to polynomials. Virtually every branch of mathematics, from algebraic number theory and algebraic geometry to applied analysis, Fourier analysis, and computer science, has its corpus of theory arising from the study of polynomials. Historically, questions relating to polynomials, for example, the solution of polynomial equations, gave rise to some of the most important problems of the day. The subject is now much too large to attempt an encyclopaedic coverage.

The body of material we choose to explore concerns primarily polynomials as they arise in analysis, and the techniques of the book are primarily analytic. While the connecting thread is the polynomial, this is an analysis book. The polynomials and rational functions we are concerned with are almost exclusively of a single variable.

We assume at most a senior undergraduate familiarity with real and complex analysis (indeed in most places much less is required). However, the material is often tersely presented, with much mathematics explored in the exercises, some of which are quite hard, many of which are supplied with copious hints, some with complete proofs. Well over half the material in the book is presented in the exercises. The reader is encouraged to at least browse through these. We have been much influenced by Pólya and Szego's classic

*Problems and Theorems in Analysis*in our approach to the exercises. (Though unlike Pólya and Szego we chose to incorporate the hints with the exercises.)

The book is mostly self-contained. The text, without the exercises, provides an introduction to the material, but much of the richness is reserved for the exercises. We have attempted to highlight the parts of the theory and the techniques we find most attractive. So, for example, Muntz's lovely characterisation of when the span of a set of monomials is dense is explored in some detail. This result epitomises the best of the subject: an attractive and nontrivial result with several attractive and nontrivial proofs.

There are excellent books on orthogonal polynomials, Chebyshev polynomials, Chebyshev systems, and the geometry of polynomials, to name but a few of the topics we cover, and it is not our intent to rewrite any of these. Of necessity and taste, some of this material is presented, and we have attempted to provide some access to these bodies of mathematics. Much of the material in the later chapters is recent and cannot be found in book form elsewhere.

**3.3. From Introduction and Basic Properties.**

The most basic and important theorem concerning polynomials is the Fundamental Theorem of Algebra. This theorem, which tells us that every polynomial factors completely over the complex numbers, is the starting point for this book. Some of the intricate relationships between the location of the zeros of a polynomial and its coefficients are explored in Section 2. The equally intricate relationships between the zeros of a polynomial and the zeros of its derivative or integral are the subject of Section 1.3. This chapter serves as a general introduction to the body of theory known as the geometry of polynomials. Highlights of this chapter include the Fundamental Theorem of Algebra, the Eneström-Kakeya theorem, Lucas' theorem, and Walsh's two-circle theorem.

**3.4. Review by: Walter Van Assche.**

*SIAM Review*

**38**(4) (1996), 705-706.

Of all the functions available for scientific work, polynomials often are considered the most friendly functions. They are easy to differentiate, have a nice closed analytic form, can be factored into a product of simpler polynomials, and the fundamental theorem of algebra tells us that any polynomial of degree

*n*has precisely

*n*complex zeros. These zeros often have a relevant meaning in various applications. Furthermore, the Weierstrass approximation theorem tells us that every continuous function on a finite interval can be approximated by polynomials; hence in various applications we can replace a given continuous function by its polynomial approximation. So polynomials really deserve special attention and the present book on polynomials win certainly find a broad readership.

This book by Borwein and Erdelyi gives an interesting approach to polynomials, with emphasis in the later chapters on polynomial inequalities. Only polynomials of one variable are treated, so those interested in polynomials in several variables will be disappointed. Not only polynomials are treated in this book: the authors observe that many properties of polynomials also hold for any Chebyshev system.

...

More emphasis in the book is placed on theory than on application. The presentation of the results is rather unusual. Often the text gives an introduction to a particular aspect of the theory, and then much more is given in the form of exercises.

...

This is a wonderful book which is strongly recommended for use in a class with students who are willing to work on the proofs, rather than to digest fully prepared and worked-out proofs and examples.

**3.5. Review by: D S Lubinsky.**

*Mathematical Reviews*MR1367960

**(97e:41001)**.

Congratulations to the authors on producing an excellent companion to G Pólya and G Szegö's classic

*Problems and theorems in analysis*. As the title suggests, its focus is narrower than that of Pólya and Szegö, concentrating on polynomials, rational functions, trigonometric polynomials, and Müntz polynomials/rational functions, but its lively tutorial style makes it a delight to read. It highlights not only the authors' recent exciting results, but also many recent works that have not been surveyed anywhere in book form, while not neglecting the classics. Chebyshev, Remez, Bernstein, Cartan, Markov, Erdös, Turán, among others, are all represented, all with new twists, as are sharp inequalities for derivatives of polynomials, trigonometric polynomials, Müntz polynomials, rationals with pre-assigned poles, polynomials with zeros in some region, etc. There are delightful applications such as the length of a lemniscate, density of weighted polynomials, incomplete polynomials, irrationality and transcendence, etc.

This reviewer will make sure to have two copies; one for reference at work, and another at home to browse, and for bedtime/recreational reading, on the shelf next to Pólya and Szegö. For those analysts who do not have polynomials on their minds, the book will be a wonderful source of useful inequalities, results and extensive references.

**4. Pi: A Source Book (1997), by Lennart Berggren, Jonathan M Borwein and Peter B Borwein.**

**4.1. Note.**

Second Edition 2000. Third Edition, incorporating A Pamphlet on Pi 2003. Fourth edition 2011.

**4.2. Preface to the First Edition.**

Our intention in this collection is to provide, largely through original writings, an extended account of

*π*from the dawn of mathematical time to the present. The story of

*π*reflects the most seminal, the most serious, and sometimes the most whimsical aspects of mathematics. A surprising amount of the most important mathematics and a significant number of the most important mathematicians have contributed to its unfolding - directly or otherwise.

Pi is one of the few mathematical concepts whose mention evokes a response of recognition and interest in those not concerned professionally with the subject, It has been a part of human culture and the educated imagination for more than twenty-five hundred years. The computation of

*π*is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modem mathematical research. To pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject that provides mathematicians with examples of many current mathematical techniques as well as a palpable sense of their historical development.

**4.3. From the publisher of the Second Edition.**

Our intention in this collection is to provide, largely through original writings, an extended account of

*π*from the dawn of mathematical time to the present. The story of

*π*reflects the most seminal, the most serious, and sometimes the most whimsical aspects of mathematics. A surprising amount of the most important mathematics and a significant number of the most important mathematicians have contributed to its unfolding directly or otherwise.

*π*is one of the few mathematical concepts whose mention evokes a response of recognition and interest in those not concerned professionally with the subject. It has been a part of human culture and the educated imagination for more than twenty-five hundred years. The computation of

*π*is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research. To pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject that provides mathematicians with examples of many current mathematical techniques as well as a palpable sense of their historical development. Why a Source Book? Few books serve wider potential audiences than does a source book. To our knowledge, there is at present no easy access to the bulk of the material we have collected.

**4.4. Preface to the Second Edition.**

We are gratified that the first edition was sufficiently well received so as to merit a second. In addition to correcting a few minor infelicities, we have taken the opportunity to add an Appendix in which articles 9 and 12 by Viète and Huygens respectively are translated into English. While modern European languages are accessible to our full community - at least through colleagues - this is no longer true of Latin. Thus, following the suggestions of a reviewer of the first edition we have opted to provide a serviceable if fairly literal translation of three extended Latin excerpts. And in particular to make Viète's opinions and style known to a broader community.

We also record that in the last two years distributed computations have been made of the binary digits of

*π*using an enhancement due to Fabrice Bellard of the identity made in article 70. In particular the binary digits of

*π*starting at the 40 trillionth place are 0 0000 1111 1001 1111. Details of such ongoing computations, led by Colin Percival, are to be found at www.cecm.sfu.ca/projects/pihex.

Corresponding details of a billion ($2^{30}$) digit computation on a single Pentium Il PC, by Dominique Delande using Carey Bloodworth's desktop

*π*program and taking under nine days, are lodged at www.cecm.sfu.ca/personal/jborwein/pi_cover.html. Here also are details of the computation of $2^{36}$ digits by Kanada et al. in April 1999.

We are grateful for the opportunity to thank Jen Chang for all her assistance with the cover design of the book. We also wish to thank Annie Marquis and Judith Borwein for their substantial help with the translated material.

**4.5. From the Publisher of Third Edition.**

This book documents the history of

*π*from the dawn of mathematical time to the present. One of the beauties of the literature on

*π*is that it allows for the inclusion of very modern, yet accessible, mathematics. The articles on

*π*collected herein fall into various classes. First and foremost there is a selection from the mathematical and computational literature of four millennia. There is also a variety of historical studies on the cultural significance of the number. Additionally, there is a selection of pieces that are anecdotal, fanciful, or simply amusing.

For this new edition, the authors have updated the original material while adding new material of historical and cultural interest. There is a substantial exposition of the recent history of the computation of digits of

*π*, a discussion of the normality of the distribution of the digits, and new translations of works by Viète and Huygens.

**4.6. Preface to the Third Edition.**

Our aim in preparing this edition is to bring the material in the collection of papers in the second edition of this source book up to date. Moreover, several delightful pieces became available and are added.

This substantial supplement to the third edition serves as a stand-alone exposition of the recent history of the computation of digits of

*π*. It also includes a discussion of the thorny old question of normality of the distribution of the digits. Additional material of historical and cultural interest is included, the most notable being new translations of the two Latin pieces of Viète (translation of article 9 (Excerpt 1): Various Responses on Mathematical Matters: Book VII (1593) and (Excerpt 2): Defense for the New Cyclometry or "Anti-Axe"), and a thorough revision of the translation of Huygens's piece (article 12) published in the second edition.

We should like to thank Professor Marinus Taisbak of Copenhagen for grappling with Viète's idiosyncratic style to produce the new translation of his work. We should like to thank Karen Aardal for permission to use her photograph of Ludolph's new tombstone in the Pieterskerk in Leiden, the Smithsonian Institution for permission to reproduce a fine photo of ENIAC, and David and Gregory Chudnovsky for providing a

*Walk on the digits of Pi*. We should also like to thank Irving Kaplansky for his gracious permission to include his

*A Song about Pi*. Finally, our thanks go to our colleagues whose continued interest in

*π*has encouraged our publisher to produce this third edition, as well as for the comments and corrections to earlier editions that some of them have sent us.

**4.7. Review by: E J Barbeau.**

*Mathematical Review*s MR1467531

**(98f:01001)**.

These 70 facsimile papers and excerpts constitute a mixture of historical tracts and expository accounts in the long, fascinating and occasionally whimsical saga of the number

*π*. Early work of ancient non-European civilisations is sampled along with European research since the seventeenth century (for example, the transcendence papers of Hermite and Lindemann). Several recent contributions record progress in computing

*π*to an incredible degree of accuracy, reflecting the expertise of two of the editors in this area. In the body of the book, the papers follow one another chronologically without explanation. For context and sources, the reader must check through the table of contents, the bibliography, the list of credits and the first appendix describing the early history of

*π*. A second appendix dates the progress in computing

*π*, while the third provides a list of formulae. The judicious representative selection makes this a useful addition to one's library as a reference book, an enjoyable survey of developments and a source of elegant and deep mathematics of different eras.

**5. Computational Excursions in Analysis and Number Theory (2002), by Peter Borwein.**

**5.1. From the Preface.**

This book is designed for a topics course in computational number theory. It is based around a number of difficult old problems that live at the interface of analysis and number theory. Some of these problems are the following:

**The Integer Chebyshev Problem.**Find a nonzero polynomial of degree

*n*with integer coefficients that has smallest possible supremum norm on the unit interval.

**Littlewood's Problem.**Find a polynomial of degree

*n*with coefficients in the set {+1, -1} that has smallest possible supremum norm on the unit disk.

**The Prouhet-Tarry-Escott Problem.**Find a polynomial with integer coefficients that is divisible by $(z - 1)^{n}$ and has smallest possible $l_{1}$ norm. (That is, the sum of the absolute values of the coefficients is minimal.)

**Lehmer's Problem.**Show that any monic polynomial $p, p(0) ≠ 0$, with integer coefficients that is irreducible and that is not a cyclotomic polynomial has Mahler measure at least 1.1762... .

All of the above problems are at least forty years old; all are presumably very hard, certainly none are completely solved; and all lend themselves to extensive computational explorations.

The techniques for tackling these problems are various and include probabilistic methods, combinatorial methods, "the circle method," and Diophantine and analytic techniques. Computationally, the main tool is the LLL algorithm for finding small vectors in a lattice.

The book is intended as an introduction to a diverse collection of techniques. For all chapters we have suggested related research papers where additional details may be pursued. There are many exercises and open research problems included. Indeed, the primary aim of the book is to tempt the able reader into the rich open possibilities for research in this area.

**5.2. Abstract.**

This book focuses on a variety of old problems in number theory and analysis. The problems concern polynomials with integer coefficients and typically ask something about the size of the polynomial with an appropriate measure of size and often with some restriction on the height and the degree.

So, for example, we might seek to minimise the supremum norm of a polynomial with integer coefficients of degree n on the unit interval. Or we might try to minimise the supremum norm on the unit disk of a polynomial all of whose coefficients are either 1 or -1. Both of these are "old plums." The first is due to Hilbert, and the second is due to Littlewood. Both problems arise in various contexts. The first gives easy Chebyshev estimates on the density of primes. The second arises in signal processing.

As is typical of these and the other problems we consider, the objects of study are very familiar. The problems are, by and large, easy to formulate, and while none have been completely solved, all have had significant progress made on them.

The tools of attack are diverse and include Diophantine, analytic, and probabilistic methods. The problems lend themselves to extensive computational exploration, and this is one of the unifying threads of this work.

No attempt is made to discuss the material in great generality; indeed, some effort is made to choose accessible special cases. Another unifying theme is that all the problems can be reformulated as problems about polynomials with integer coefficients, even though they often arise in other contexts.

**5.3. Review by: Sergei V Konyagin.**

*Mathematical Reviews*MR1912495

**(2003m:11045)**.

The author gives a wonderful overview of one of the most beautiful and active areas of current computational number theory. ... In the Introduction the author gives a list of problems discussed in the book. ... The problems have been investigated by many outstanding mathematicians. Most of the problems have resisted solution for at least fifty years. The author gives an overview of the contemporary progress in the problems and presents the main ideas of the proofs. The book contains necessary background, many exercises and research problems.

In conclusion, I would like to stress that the present book is on the whole very clearly written. The interested reader may also consult the numerous bibliographic references given at the end.

**6. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike (2007), by Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller.**

**6.1. From the Publisher.**

This book presents the Riemann Hypothesis, connected problems, and a taste of the body of theory developed towards its solution. It is targeted at the educated non-expert. Almost all the material is accessible to any senior mathematics student, and much is accessible to anyone with some university mathematics. The appendices include a selection of original papers that encompass the most important milestones in the evolution of theory connected to the Riemann Hypothesis. The appendices also include some authoritative expository papers. These are the "expert witnesses" whose insight into this field is both invaluable and irreplaceable.

**6.2. From the Back Cover.**

The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as "the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors."

The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers.

This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis.

The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.

**6.3. From the Preface.**

Since its inclusion as a Millennium Problem, numerous books have been written to introduce the Riemann hypothesis to the general public. In an average local bookstore, it is possible to see titles such as John Derbyshire's

*Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*, Dan Rockmore's

*Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers*, and Karl Sabbagh's

*The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics*.

This book is a more advanced introduction to the theory surrounding the Riemann hypothesis. It is a source book, primarily composed of relevant original papers, but also contains a collection of significant results. The text is suitable for a graduate course or seminar, or simply as a reference for anyone interested in this extraordinary conjecture.

The material in Part I (Chapters 1-10) is mostly organised into independent chapters and one can cover the material in many ways. One possibility is to jump to Part II and start with the four expository papers in Chapter 11. The reader who is unfamiliar with the basic theory and algorithms used in the study of the Riemann zeta function may wish to begin with Chapters 2 and 3. The remaining chapters stand on their own and can be covered in any order the reader fancies (obviously with our preference being first to last). We have tried to link the material to the original papers in order to facilitate more in-depth study of the topics presented.

We have divided Part II into two chapters. Chapter 11 consists of four expository papers on the Riemann hypothesis, while Chapter 12 contains the original papers that developed the theory surrounding the Riemann hypothesis.

Presumably the Riemann hypothesis is very difficult, and perhaps none of the current approaches will bear fruit. This makes selecting appropriate papers problematical. There is simply a lack of profound developments and attacks on the full problem. However, there is an intimate connection between the prime number theorem and the Riemann hypothesis. They are connected theoretically and historically, and the Riemann hypothesis may be thought of as a grand generalisation of the prime number theorem. There is a large body of theory on the prime number theorem and a progression of solutions. Thus we have chosen various papers that give proofs of the prime number theorem.

While there have been no successful attacks on the Riemann hypothesis, a significant body of evidence has been generated in its support. This evidence is often computational; hence we have included several papers that focus on, or use computation of, the Riemann zeta function. We have also included Weil's proof of the Riemann hypothesis for function fields and the deterministic polynomial primality test of Argawal at al.

**7. Selected Writings on Experimental and Computational Mathematics (2010), by Jonathan M Borwein and Peter B Borwein.**

**7.1. From the Publisher.**

A quiet revolution in mathematical computing and scientific visualisation took place in the latter half of the 20th century. These developments have dramatically enhanced modes of mathematical insight and opportunities for "exploratory" computational experimentation. This volume collects the experimental and computational contributions of Jonathan and Peter Borwein over the past quarter century.

**7.2. From the Preface.**

This is a representative collection of most of our writings about the nature of mathematics over the past twenty-five years. Many are jointly authored, others are by one or other of us with various co-authors - often with our long-time collaborator and friend David Bailey who has generously written a foreword to the volume. We have included some technical papers only when they form the basis for discussion in the more general papers collected. We would like to thank Joshua Borwein for writing the index for our book.

For each article in this selection, we have written a few paragraphs of introduction situating the given paper in the collection and where appropriate bringing it up to date. For the most part, we let the articles speak for themselves and suggest the reader consult the index which we have added to improve navigation between selections.

**7.3. Foreword by David Bailey.**

Great things often begin in humble ways.

My 25-year-long collaboration with Jonathan and Peter Borwein began in 1985, when I read their paper on fast methods for computing elementary constants and functions, which article is included as the first chapter of this volume. After reading this engaging and intriguing article, I was inspired to try to implement some of these algorithms on the computer, using a high-precision arithmetic software package that I wrote specifically for this task. Indeed, the Borwein algorithms worked as advertised, yielding many thousands of correct digits in just a few seconds on a state-of-the-art computer at the time. I was particularly intrigued by the Borwein algorithm for

*π*. I then contacted the Borweins to tell of my interest in their work, and, as they say, the rest is history.

As can be readily seen in the articles here, the Borweins are inarguably the world's leading exponents of utilising state-of-the-art computer technology to discover and prove new and fundamental mathematical results. They employ raw numerical computation, symbolic processing and advanced visualisation facilities in their work. The Borweins have spread this gospel in countless fascinating and cogent lectures, as well as in numerous books and over 200 published papers. As a direct result of the Borweins' influence, hundreds of researchers worldwide are now engaged in "experimental" and computationally-assisted mathematics, and the pace of mathematical discovery has measurably quickened.

In reading through these papers, I am struck that the Borweins have also made an important contribution to a more fundamental problem in the field: How can researchers, labouring at the state of the art in the very difficult and demanding arena of modern mathematics, communicate the excitement of their work to young minds who potentially will form the next generation of mathematicians?

The Borweins have found the answer: (a) Bring computers into all aspects of the mathematical research arena, thereby attracting thousands of young, computer-savvy students into the fold, and letting them experience first-hand the excitement of discovering heretofore unknown facts of mathematics; and (b) Highlight the numerous intriguing connections of this work to other fields of mathematics, computer science and modern scientific philosophy.

These articles exemplify the exploratory spirit of truly pioneering work. They are destined to be read over and over again for decades to come. What's more, in most cases these papers can be read and comprehended even by persons of modest mathematical training. Enjoy!

**8. Mathematicians on Creativity, edited by Peter Borwein, Peter Liljedahl and Helen Zhai.**

**8.1. From the Back Cover.**

This book aims to shine a light on some of the issues of mathematical creativity. It is neither a philosophical treatise nor the presentation of experimental results, but a compilation of reflections from top-calibre working mathematicians. In their own words, they discuss the art and practice of their work. This approach highlights creative components of the field, illustrates the dramatic variation by individual, and hopes to express the vibrancy of creative minds at work. Mathematicians on Creativity is meant for a general audience and is probably best read by browsing.

**8.2. From the Publisher.**

In their own words, many of the world's foremost mathematicians discuss the art and practice of their work in this book, which shines a light on some of the issues of mathematical creativity. It is neither a philosophical treatise nor the presentation of experimental results, but a compilation of reflections from top-calibre working mathematicians. This approach highlights the creative aspects of the ﬁeld, illustrates the dramatic variation by individual, and hopes to express the vibrancy of creative minds at work. Organised alphabetically, this book is meant for a general audience and is best read by browsing. It can be used as a supplementary text in mathematics history and mathematics education courses.

**8.3. Review by Ad Meskens.**

*Mathematical Reviews*MR3223097.

This is not a page-turner; it is a book which needs to be read in bits and pieces. This is a book about how mathematicians see their profession, or better still it is the mathematicians themselves who give their -sometimes outspoken - view of their subject. Sometimes witty, sometimes dull, always surprising.

The best way to read this book is first to leaf through it, then put it next to your armchair and every once in a while read one of the comments.

When you're dealing with a mathematical problem you can find some inspiration in it, not just for the problem at hand, but in general, by philosophising on what you are actually doing. Surely you will agree that "mathematical work does not proceed along the narrow logical path of truth to truth to truth, but bravely or gropingly follows deviations through the surrounding marshland of propositions" or that "to find oneself lost in wonder at some manifestation is frequently the half of a new discovery."

This book is recommended for all those who wish to know the deeper thoughts of their colleagues.

**8.4. Review by: Robert E O'Malley Jr.**

*SIAM Review*

**57**(1) (2015), 164.

This publication is a result of a survey of prominent mathematicians about the process of mathematical discovery conducted early in this century, supplemented by material taken from sources such as

*Mathematical People*and popular biographies and articles. Survey responses are published from twenty-five or so mathematicians, including E Bombieri, D Donoho, G Faltings, C Fefferman, P Huber, D Kleitman, J Marsden, G Papanicolaou, J Taylor, and G Wahba. Predictably, they don't all quite agree. For example, C Peskin wrote, "I do my best work while asleep," and A Weinstein wrote, "Most of what I 'discover' in my dreams turns out to be nonsense." The extent of nonoriginal material might correspond to the A listing: Aitkin, d'Alembert, Andrews, Aristotle, Askey, and Atiyah. Browsing is recommended. Read your heroes' thoughts.

**8.5. Review by: Marion Cohen.**

*The American Mathematical Monthly*

**122**(6) (2015), 613-616.

To what extent are mathematicians interested in the creative process that goes into their work? And to what extent do they want to talk about it? Of the 86 mathematicians quoted in this book, at least one (Joseph Doob) expressed a kind of disinterest: "Creative process is for the birds. ... I am not a cosmic thinker." Others made comments tantamount to: "The mathematics itself is too captivating to be self-conscious about the creative process." Still others said things like "[Mathematicians] are more interested in doing mathematics than speaking about it." (Paul Malliavin).

However, most of this book's mathematicians seem quite happy to talk about their creative processes, and it also seems they'd already given the matter some thought. Also, before publication of this book there was already a vast literature on topics related to mathematical creativity. Albers'

*Mathematical People: Profiles and Interviews*might attest to this. We might also mention Fitzgerald's

*The Mind of the Mathematician*, Halmos'

*I Want to Be a Mathematician*, Krantz's

*Mathematical Apocrypha*, as well as Hadamard's

*Essay on the Psychology of Invention in the Mathematical Field*.

This last mentioned is important because it was the main forerunner of our book in question. A secondary forerunner, as the Introduction describes, was a 1902 30-question survey, published in

*L'Enseignement Mathematique*, by two French psychologists, Claparede and Flournoy. This survey focused on reasons for becoming mathematicians, attitudes about everyday life, and hobbies. These authors followed up, in 1904, with the second half of their survey, focusing on mental images during mathematical work.

But Hadamard (a mathematician appearing in the book) felt that their treatment wasn't comprehensive enough, nor the subjects prominent enough. His essay included answers by mathematicians to 33 questions - which appear on pages 151-154 of our book. Our editors focused on five of Hadamard's questions, listed on pages xiii-xiv, such as "Would you say that your principal discoveries have been the result of deliberate endeavour in a definite direction, or have they arisen ... spontaneously?"

This book makes interesting reading, and a very enjoyable complement to the several books mentioned above and to the works listed in its Bibliography. It's nice to have in one place so many great quotes by mathematicians about their creative processes.

**8.6. Review by: Joshu Fisher.**

*The Mathematics Teacher*

**108**(9) (2015), 716.

What is the process by which mathematics is created? This collection of quotations and snippets of work offers some insight into the minds of many gifted mathematicians and how they view the creative process within their field. With ties to earlier surveys of practicing mathematicians (significantly, by Hadamard), the authors set out to combine both contemporary and historical perspectives on this issue.

*Mathematicians on Creativity*has no narrative arc or argument and little in the way of summary. Readers are encouraged to browse (rather than read straight through) and to draw their own conclusions. The result is a somewhat disjointed effect.

In the end, finding the right use for the book is somewhat challenging. As an occasional read, it may serve as a casual and enjoyable distraction. Careful reading selections could also serve well during an introduction to proof, helping show students that mathematics is both challenging and beautiful and that even the very best mathematicians sometimes need to rely on a "Eureka! moment." Although the selection and content of the included quotations are quite interesting, the book ultimately left me wanting more depth of organisation and analysis.

Last Updated September 2023