Pi: The Next Generation

Transcription

1 Pi: The Next Generation

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3 . David H. Bailey Jonathan M. Borwein Pi: The Next Generation A Sourcebook on the Recent History of Pi and Its Computation

4 David H. Bailey Lawrence Berkeley National Laboratory Berkeley, CA, USA Jonathan M. Borwein Centre for Computer Assisted Research Mathematics and its Applications, School of Mathematical and Physical Sciences University of Newcastle Newcastle, Australia ISBN DOI / ISBN (ebook) Library of Congress Control Number: Mathematics Subject Classification (2010): 01-00, 01-08, 01A05, 01A75, 11-00, 11-03, 26-03, 65-03, Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

5 To Our Grandchildren

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7 Contents Foreword Preface ix xiii 1. Computation of π using arithmetic-geometric mean (1976) 1 2. Fast multiple-precision evaluation of elementary functions (1976) 9 3. The arithmetic-geometric mean of Gauss (1984) The arithmetic-geometric mean and fast computation of elementary functions (1984) A simplified version of the fast algorithms of Brent and Salamin (1985) Is pi normal? (1985) The computation of π to 29,360,000 decimal digits using Borweins quartically convergent algorithm (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary (1988) Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of pi calculation (1988) Ramanujan and pi (1988) Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi (1989) Pi, Euler numbers, and asymptotic expansions (1989) A spigot algorithm for the digits of π (1995) On the rapid computation of various polylogarithmic constants (1997) Similarities in irrationality proofs for π, ln 2, ζ(2), and ζ(3) (2001) Unbounded spigot algorithms for the digits of pi (2006) Mathematics by experiment: Plausible reasoning in the 21st Century (2008) Approximations to π derived from integrals with nonnegative integrands (2009) Ramanujan s series for 1/π: A survey (2009) 303 vii

8 viii CONTENTS 20. The computation of previously inaccessible digits of π 2 and Catalan s constant (2013) Walking on real numbers (2013) Birth, growth and computation of pi to ten trillion digits (2013) Pi day is upon us again and we still do not know if pi is normal (2014) The Life of π (2014) I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly (2015) 475 Index 501

9 Foreword The fundamental constant π has played an indispensable role in mathematics, science, and most world cultures for over 4000 years. One would like to calculate the circumference of a circle without actually physically measuring it. The ancient Babylonians and Hebrews recognized that it was considerably easier to determine the radius of a circle than its circumference, and so the more physically demanding task of calculating the circumference could be replaced by simply multiplying the radius by 3. However, before 2000 BCE, the Babylonians replaced this initial approximation of 3 for π by On the Egyptian Rhind Papyrus from about 1650 BCE, we find the slightly better approximation 4(8/9) 2 = We have come a long way since the heydays of the Babylonian, Hebrew, and Egyptian cultures. With the calculation of houkouonchi and Alexander J. Yee, we now know 13.3 trillion digits of π. It might be pointed out that π was not always denoted by π; William Jones first used the notation π in Euler adopted the symbol by 1737, and since that time, the notation π has been universally used. The value π arises in many contexts in mathematics in surprising ways. One of these is the Buffon needle problem, which asks for the probability that a needle of length l will land on one of the vertical, equally spaced parallel lines on a floor, where the distance between each two adjacent lines is equal to d. If d = l, this probability is 2/π. Any calculus student will testify that the value of π often arises in the calculation of a definite integral or of an infinite series in closed form. The great Indian mathematician, Srinivasa Ramanujan, evaluated a plethora of integrals and infinite series in closed form, and he also found many beautiful series identities. Chapter 14 in his second notebook contains 87 such results, and of these series and integral evaluations in closed form and series identities, I counted 70 of them in which π appears. Unusual or surprising formulas for π are a source of delight for many of us. When we calculate the simple continued fraction π = , of π, we are perhaps surprised by the large denominator 293. This indicates that if we truncate the continued fraction just prior to this large denominator, we should obtain a good approximation to π. Indeed, the associated approximation gives the first six digits of π. On the other hand, examining the simple continued fraction π 4 = , for π 4, we find the huge denominator This leads to the approximations π = , π , 22 while π 4 = , π = ix

10 x FOREWORD Infinite series, perhaps beginning with the familiar Madhava Gregory series for Arctan x, have played important roles in calculating the digits of π. One of the most beautiful series representations for π is given by 9801 π 8 = n=0 (4n)!( n) (n!) 4 (396) 4n, (1) stated without proof along with 16 further such series for 1/π by Ramanujan in his famous paper, Modular equations and approximations to π. In November, 1985, R. William Gosper, Jr., employed (1) to calculate 17,526,100 digits of π, which was a world record at that time. Each term of (1) gives about 8 digits of π per term. Perhaps we should have put quotation marks about world record, because at that time no proof of (1) had ever been given. We now have proofs of (1), Ramanujan s 16 further series formulas for 1/π, and many similar series representations for 1/π, in particular, one due to David and Gregory Chudnovsky that yields 14 digits of π per term. Papers numbered 10, 11, and 19 in this volume provide readers with more details about these formulas and their use for computing digits of π. There are many further intriguing formulas for π. One of the most popular is John Wallis s formula (1655) π = My favorite is Lord Brouncker s ( ) continued fraction for π, π = (2) Brouncker was a student of Wallis, and mathematicians have speculated for many years on how Brouncker might have derived his formula (2). It is actually a special case of a more general continued fraction Γ( 1 4 (x + n + 1))Γ( 1 4 (x n + 1)) Γ( 1 4 (x + n + 3))Γ( 1 4 (x n + 3)) = 4 n n n x 2x 2x 2x, (3) which can be found as Entry 25 in Chapter 12 of Ramanujan s second notebook. If we set x = 1 and n = 0 in (3), we obtain (2). Seemingly strange approximations to π can be obtained from class invariants. Let n be a positive integer, and set q = e π n. The class invariant G n is defined by G n := 2 1/4 q 1/24 (1 + q)(1 + q 3 )(1 + q 5 ). (4) Class invariants are algebraic numbers. Thus, taking the logarithm of both sides of (4) for a certain class invariant should produce an approximate formula for π. For example, on page 300 in his second notebook, Ramanujan considered the cubic equation 2x 3 4x 2 + 6x 1 = 0, which has the real root Note that the first 16 digits of agree with the first 16 digits of x 0 = log x = π =

11 FOREWORD xi It may come as a surprise to readers that the resurgence in calculating the digits of π arose from an adaptation of the arithmetic-geometric mean of Gauss by (individually) Eugene Salamin and Richard Brent. Readers can learn about these developments in the first five papers of this volume. Although we know a lot of digits of π, there is much that we do not know about these digits. For example, we would conjecture that the average digit should be about 4.5, but we do not know how to prove this. More precisely, it is conjectured that π is normal, but evidently we are far from proving this as well. This volume contains papers by Stan Wagon (no. 6); by the present authors (no. 17); by Francisco Aragon Artacho, the present authors, and Peter Borwein (no. 21); and by the present authors (no. 23) bringing readers up to date on what we know about this famous conjecture. We have provided only a small sample of examples to illustrate the beauty and mystery of this remarkable number. Mathematicians and computer scientists continue to delve into the mysteries of π. Our goal in the present volume is to bring together some of these investigations and thoughts about π from papers published during the past half century. Many of these papers can be read by a broad audience. For readers who want to read further, most of the 25 selected papers point to more technical articles. Bruce C. Berndt

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13 Preface This volume is a companion to Pi: A Source Book (by Lennart Berggren, Jonathan Borwein, and Peter Borwein, Springer-Verlag), which was first published in 1997, with a third edition released in Rather than produce an even heftier fourth edition, the current authors have prepared a collection of papers written between 1975 and the present. Since a number of the collected papers contain substantial historical material, the reader can glean an accurate picture of the life of π from the current volume. That said, the focus in this book is on π in the digital age. The reader will note that many of the papers have substantial algorithmic material and it is recommended that where possible he or she explore such material at the computer. Each of the 25 papers comprising this volume is preceded by a brief summary of its contents, and this is accompanied by a very brief, key word, indication of some of the ways the content of the given paper relates to that of others in the collection. This information is also recorded in Table 1 below. For the most part, however, we are happy to let the papers speak for themselves. The present authors have been fascinated by π throughout their academic lives and hope that this volume will help readers share this fascination and potentially even contribute to the ever-growing literature on the subject. The associated webpage for the collection is accessible through We are also delighted that Bruce Berndt agreed to add a Foreword to the volume. Algorithms 1, 2, 4, 5, 8, 10, 11, 13, 14, 16, 17, 22, 23, 24, 25 Arithmetic-geometric mean 1, 2, 3, 4, 5, 8 Computation 1, 2, 4, 6, 7, 8, 9, 11, 14, 17, 19, 20, 22, 24, 25 Curiosities 12, 21, 22 Elliptic integrals 1, 2, 3, 8, 11 General audience 6, 10, 23, 24 Graphical representation 21 History 3, 4, 8, 10, 11, 17, 19, 20, 22, 24, 25 Approximations 2, 4, 18 Irrationality 15, 24, 25 Modular equations 10, 11 Normality 6, 7, 9, 17, 20, 21, 23 Random walks 21 Series 12, 19, 22 Table 1. Articles by Keyword xiii

14 xiv PREFACE We wish to thank visiting Waterloo students Sally Dong and Jason Lynch for their help, Dr. Daniel Sutherland for his substantial assistance, and our Springer- Verlag editor Elizabeth Loew for her active encouragement to produce this volume. Finally, we thank the original publishers for their permissions to republish the articles herein. In particular, we would like to thank: ACM, for permission to reprint paper 2 AMS, for permission to reprint papers 1, 5, 7, 14, and 20 ICMI, for permission to reprint paper 3 IEEE, for permission to reprint paper 9 MAA, for permission to reprint 8, 11, 12, 13, 15, 16, 18, 19, 23, and 25 SIAM, for permission to reprint paper 4 Springer-Verlag, for permission to reprint papers 6, 21, and 24 Taylor & Francis, for permission to reprint entry 17 Berkeley, CA, USA Newcastle, NSW, Australia March 2016 David H. Bailey Jonathan M. Borwein