Numerical Algorithms for Number Theory: Using Pari/GP (Mathematical Surveys and Monographs, 254)

442 Pages·2021·2.144 MB·English

by  Karim Belabas, Henri Cohen

Mathematical
Surveys
and
Monographs
Volume 254
Numerical Algorithms
for Number Theory
Using Pari/GP
Karim Belabas
Henri Cohen
Numerical Algorithms
for Number Theory
Using Pari/GP

Mathematical
Surveys
and
Monographs
Volume 254
Numerical Algorithms
for Number Theory
Using Pari/GP
Karim Belabas
Henri Cohen
EDITORIAL COMMITTEE
Ana Caraiani Natasa Sesum
Robert Guralnick, Chair ConstantinTeleman
Bryna Kra Anna-Karin Tornberg
2020 Mathematics Subject Classification. Primary 11-04, 11Y60, 65Y20, 11M06, 11M41,
30B70, 65B10, 65D30.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-254
Library of Congress Cataloging-in-Publication Data
Names: Belabas,Karim,author. |Cohen,Henri,author.
Title: Numericalalgorithmsfornumbertheory: usingPari/GP|KarimBelabas,HenriCohen.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2021]|Series: Mathe-
maticalsurveysandmonographs,0076-5376;volume254|Includesbibliographicalreferences
andindex.
Identifiers: LCCN2021007368|ISBN9781470463519(paperback)|9781470465568(ebook)
Subjects: LCSH: Numerical analysis–Computer programs. | Number theory. | Computer algo-
rithms. |AMS:Numbertheory–Software,sourcecode,etc. forproblemspertainingtonumber
theory. | Number theory – Computational number theory – Evaluation of number-theoretic
constants. | Numerical analysis – Computer aspects of numerical algorithms – Complexity
and performanceofnumericalalgorithms. | Number theory– Zeta and L-functions: analytic
theory; ζ(s) and L(s,χ). | Number theory – Zeta and L-functions: analytic theory – Other
Dirichlet series and zeta functions. | Functions of a complex variable – Series expansions of
functionsofonecomplexvariable–Continuedfractions;complex-analyticaspects. |Numerical
analysis–Accelerationofconvergenceinnumericalanalysis–Numericalsummationofseries.
|Numericalanalysis–Numericalapproximationandcomputationalgeometry(primarilyalgo-
rithms)–Numericalintegration.
Classification: LCCQA297.B3752021|DDC518/.47–dc23
LCrecordavailableathttps://lccn.loc.gov/2021007368
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Contents
Preface ix
Chapter 1. Introduction 1
1.1. Subject matter 1
1.2. Experimental protocols 2
1.3. Multiprecision algorithms and working accuracy 2
1.4. Comments on the GP language 3
1.5. Warnings 7
1.6. Examples 8
Chapter 2. Numerical extrapolation 11
2.1. Introduction 11
2.2. Richardson extrapolation 14
2.3. Interlude: Estimating the number N of nodes 17
2.4. Extrapolating by interpolation: Lagrange 22
2.5. Extrapolation using Sidi’s mW algorithm 34
2.6. Computing asymptotic expansions 37
2.7. Sample timings for Limit programs 43
2.8. Conclusion 48
Chapter 3. Numerical integration 51
3.1. Numerical differentiation 51
3.2. Integration of rational functions 57
3.3. Generalities on numerical integration 65
3.4. Newton–Cotes type methods 73
3.5. Orthogonal polynomials 81
3.6. Gaussian integration methods 99
3.7. Gaussian Integration on [a,∞] 121
3.8. Doubly-exponential integration methods (DE) 126
3.9. Integration of oscillatory functions 138
3.10. Sample timings for integrals on [a,b] 154
3.11. Sample timings for integrals on [0,∞] 157
3.12. Sample timings for oscillatory integrals 162
3.13. Final conclusion on numerical integration 164
Chapter 4. Numerical summation 165
4.1. Introduction 165
4.2. Euler–Maclaurin summation methods 166
4.3. Pinelis summation 201
4.4. Sums and products of rational functions 204
v
vi CONTENTS
4.5. Summation of oscillating series 206
4.6. Summing by extrapolation 219
4.7. Van Wijngaarden’s method 223
4.8. Monien summation 224
4.9. Summing functions defined only on integers 235
4.10. Multiple sums and multizeta values 236
4.11. Sample timings for summation programs 245
4.12. Sample timings for Sumalt programs 249
Chapter 5. Euler products and Euler sums 253
5.1. Euler sums 253
5.2. Euler products 256
5.3. Variants involving log(p) or log(log(p)) 257
5.4. Variants involving quadratic characters 260
5.5. Variants involving congruences 262
5.6. Hardy–Littlewood constants: Quadratic polynomials 263
5.7. Hardy–Littlewood constants: General polynomials 267
Chapter 6. Gauss and Jacobi sums 273
6.1. Gauss and Jacobi sums over F 273
q
6.2. Practical computations of Gauss and Jacobi sums 278
6.3. Using the Gross–Koblitz formula 284
6.4. Gauss and Jacobi sums over Z/NZ 290
Chapter 7. Numerical computation of continued fractions 297
7.1. Generalities 297
7.2. Na¨ıve numerical computation 298
7.3. Speed of convergence of infinite continued fractions 300
7.4. Examples of each convergence case 310
7.5. Convergence acceleration of continued fractions 319
7.6. The quotient-difference algorithm 324
7.7. Evaluation of the quotient-difference output 327
Chapter 8. Computation of inverse Mellin transforms 331
8.1. Introduction 331
8.2. Gamma products 332
8.3. Compendium of possible methods 335
8.4. Using the power series around x=0 336
8.5. Using the asymptotic expansion 338
8.6. Generalized incomplete Gamma functions 343
Chapter 9. Computation of L-functions 345
9.1. The basic setting and goals 345
9.2. The associated Theta function 346
9.3. Computing Λ(s) and L(s) 357
9.4. Booker–Molin’s idea for computing Λ(s): Poisson summation 360
9.5. The Fourier error 363
9.6. The truncation errors 365
9.7. Implementation 368
9.8. A possible program for computing Λ(s) and L(s) 371
CONTENTS vii
9.9. Applications 374
9.10. Examples 379
9.11. Shifting the line of integration in the Booker–Molin method 386
9.12. Computing L(s) for large (cid:3)(s) 387
9.13. Explicit formulas 403
Appendix A. List of relevant GP programs 415
Bibliography 417
Index of Programs 421
General Index 425

Preface
Thisbookisalargelyexpandedversionofacoursethatthesecondauthorgave
atICTPinTriesteinthesummer2012,precedingaconferenceon“hypergeometric
motives”,inRennesinApril2014attheJourn´eesLouisAntoine, andattheNTCS
workshop in Taiwan in August 2014.
The goalofthisbookistopresent anumber ofanalyticandarithmeticnumer-
ical methods used in number theory, with a particular emphasis on the ones which
are less known than they should be, although very classical tools are also men-
tioned. Note that, as is very often the case in number theory, we want numerical
methodsgivingsometimes hundreds ifnotthousandsofdecimal placesofaccuracy.
The typical timing tables that we will give are in fact for 500 decimal digits.
The style of presentation is the following: we first give proofs of some of the
tools, the prerequisites being classical undergraduate analysis. Note that since
the emphasis is on practicality, the proofs are sometimes only heuristic, but valid
in actual practice. We then give the corresponding Pari/GP programs, usually
followed by a number of examples. These programs are also available as a unique
separate archive on the authors’ website at
http://www.math.u-bordeaux.fr/~kbelabas/Numerical_Algorithms/
Feelfreetoexperimentandmodifythemtoyourheart’scontent. Theycanalso
serve as an introduction to the syntax and semantics of GP, since in general they
are easy to understand and do not use much sophistication. Note we use rather
recent features of the GP language, so we strongly advise to download the latest
release (version 2.13 or more recent) from the Pari/GP website:
http://pari.math.u-bordeaux.fr/
The reader is advised to refer to the numerous books dealing with parts of the
subject, such as (but of course not limited to!) the second author’s four books
[Coh93], [Coh00], [Coh07a], and [Coh07b].
Caveat: Neither the authors nor the AMS are liable for any damage caused
by the use of the programs given in this book. Apart from this legalese, we would
be happy to hear of any corrections and/or improvements. Even though we have
triedtogivesimpleandefficientprograms, wedonotclaimthatwegive“thebest”
methods, and we would also be very glad to hear of new methods for solving the
problems considered in this book.
Karim Belabas and Henri Cohen
ix