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Numerical Algorithms for Number Theory: Using Pari/GP
About this Title
Karim Belabas, Université de Bordeaux, Bordeaux, France and Henri Cohen, Université de Bordeaux, Bordeaux, France
Publication: Mathematical Surveys and Monographs
Publication Year:
2021; Volume 254
ISBNs: 978-1-4704-6351-9 (print); 978-1-4704-6556-8 (online)
DOI: https://doi.org/10.1090/surv/254
ReadershipTable of Contents
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Front/Back Matter
Chapters
- Introduction
- Numerical extrapolation
- Numerical integration
- Numerical summation
- Euler products and Euler sums
- Gauss and Jacobi sums
- Numerical computation of continued fractions
- Computation of inverse Mellin transforms
- Computation of $L$-functions
- List of relevant GP programs
- J. Arias de Reyna, High precision computation of Riemann’s zeta function by the Riemann-Siegel formula, I, Math. Comp. 80 (2011), no. 274, 995–1009. MR 2772105, DOI 10.1090/S0025-5718-2010-02426-3
- P. Akhilesh, Double tails of multiple zeta values, J. Number Theory 170 (2017), 228–249. MR 3541706, DOI 10.1016/j.jnt.2016.06.020
- Frits Beukers, Henri Cohen, and Anton Mellit, Finite hypergeometric functions, Pure Appl. Math. Q. 11 (2015), no. 4, 559–589. MR 3613122, DOI 10.4310/PAMQ.2015.v11.n4.a2
- Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1625181
- Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis, Experiment. Math. 15 (2006), no. 4, 385–407. MR 2293591
- N. Bourbaki, Éléments de mathématique. IX. Première partie: Les structures fondamentales de l’analyse. Livre IV: Fonctions d’une variable réelle (théorie élémentaire). Chapitre I: Dérivées. Chapitre II: Primitives et intégrales. Chapitre III: Fonctions élémentaires, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1074, Hermann & Cie, Paris, 1949 (French). MR 0031013
- Richard Brent, David Platt, and Timothy Trudgian, Accurate estimation of sums over zeros of the riemann zeta-function, 2020, preprint, arXiv:2009.13791.
- Richard P. Brent and Paul Zimmermann, Modern computer arithmetic, Cambridge Monographs on Applied and Computational Mathematics, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2760886
- Edgar Costa, Kiran Kedlaya, and David Roe, Hypergeometric $L$-functions in average polynomial time, 2020, preprint, arXiv:2005.13640.
- Henri Cohen, Exponential sums can (usually) be computed explicitly, preprint.
- Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
- Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
- Henri Cohen, Number theory. Vol. I. Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, Springer, New York, 2007. MR 2312337
- Henri Cohen, Number theory. Vol. II. Analytic and modern tools, Graduate Texts in Mathematics, vol. 240, Springer, New York, 2007. MR 2312338
- Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence acceleration of alternating series, Experiment. Math. 9 (2000), no. 1, 3–12. MR 1758796
- Henri Cohen and Fredrik Strömberg, Modular forms, Graduate Studies in Mathematics, vol. 179, American Mathematical Society, Providence, RI, 2017. A classical approach. MR 3675870, DOI 10.1090/gsm/179
- Henri Cohen and Don Zagier, Vanishing and non-vanishing theta values, Ann. Math. Qué. 37 (2013), no. 1, 45–61 (English, with English and French summaries). MR 3117737, DOI 10.1007/s40316-013-0003-x
- Tim Dokchitser, Computing special values of motivic $L$-functions, Experiment. Math. 13 (2004), no. 2, 137–149. MR 2068888
- Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, 2019, preprint, arXiv:1908.06808.
- Kurt Fischer, The Zetafast algorithm for computing zeta functions, 2017, preprint, arXiv:1703.01414v7.
- William F. Ford and Avram Sidi, An algorithm for a generalization of the Richardson extrapolation process, SIAM J. Numer. Anal. 24 (1987), no. 5, 1212–1232. MR 909075, DOI 10.1137/0724080
- Xavier Gourdon, Algorithmique du théorème fondamental de l’algèbre, Rapport de recherche 1852, INRIA, 1993.
- E. Hecke, Über analytische Funktionen und die Verteilung von Zahlen mod. eins, Abh. Math. Sem. Univ. Hamburg 1 (1922), no. 1, 54–76 (German). MR 3069388, DOI 10.1007/BF02940580
- Peter Henrici, The quotient-difference algorithm, Nat. Bur. Standards Appl. Math. Ser. 49 (1958), 23–46. MR 94901
- Ghaith A. Hiary, Computing Dirichlet character sums to a power-full modulus, J. Number Theory 140 (2014), 122–146. MR 3181649, DOI 10.1016/j.jnt.2013.12.005
- Fredrik Johansson and Marc Mezzarobba, Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights, SIAM J. Sci. Comput. 40 (2018), no. 6, C726–C747. MR 3880259, DOI 10.1137/18M1170133
- Fredrik Johansson, Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, Numer. Algorithms 69 (2015), no. 2, 253–270. MR 3350381, DOI 10.1007/s11075-014-9893-1
- Fredrik Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66 (2017), no. 8, 1281–1292. MR 3681746, DOI 10.1109/TC.2017.2690633
- Jerzy Kaczorowski, Axiomatic theory of $L$-functions: the Selberg class, Analytic number theory, Lecture Notes in Math., vol. 1891, Springer, Berlin, 2006, pp. 133–209. MR 2277660, DOI 10.1007/978-3-540-36364-4_{4}
- M. Kaneko and D. Zagier, Supersingular $j$-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, Computational perspectives on number theory (Chicago, IL, 1995) AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, 1998, pp. 97–126. MR 1486833, DOI 10.1090/amsip/007/05
- David Levin, Development of non-linear transformations of improving convergence of sequences, Internat. J. Comput. Math. 3 (1973), 371–388. MR 359261, DOI 10.1080/00207167308803075
- Stéphane R. Louboutin, Efficient computation of class numbers of real abelian number fields, Algorithmic number theory (Sydney, 2002) Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 134–147. MR 2041079, DOI 10.1007/3-540-45455-1_{1}1
- S. K. Lucas and H. A. Stone, Evaluating infinite integrals involving Bessel functions of arbitrary order, J. Comput. Appl. Math. 64 (1995), no. 3, 217–231. MR 1365426, DOI 10.1016/0377-0427(95)00142-5
- Yudell L. Luke, The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0241700
- Jean-François Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Math. 58 (1986), no. 2, 209–232 (French). MR 844410
- Pascal Molin, Intégration numérique et calculs de fonctions $L$, Ph.D. thesis, Université Bordeaux 1, 2010.
- H. Monien, Gaussian quadrature for sums: a rapidly convergent summation scheme, Math. Comp. 79 (2010), no. 270, 857–869. MR 2600547, DOI 10.1090/S0025-5718-09-02289-3
- Takuya Ooura and Masatake Mori, The double exponential formula for oscillatory functions over the half infinite interval., J. Comput. Appl. Math. 38 (1991), no. 1-3, 353–360 (English).
- Takuya Ooura and Masatake Mori, A robust double exponential formula for Fourier-type integrals, J. Comput. Appl. Math. 112 (1999), no. 1-2, 229–241. Numerical evaluation of integrals. MR 1728462, DOI 10.1016/S0377-0427(99)00223-X
- Iosif Pinelis, An alternative to the Euler-Maclaurin summation formula: approximating sums by integrals only, Numer. Math. 140 (2018), no. 3, 755–790. MR 3854359, DOI 10.1007/s00211-018-0978-y
- William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical recipes, 3rd ed., Cambridge University Press, Cambridge, 2007. The art of scientific computing. MR 2371990
- Olivier Ramaré and Aled Walker, Products of primes in arithmetic progressions: a footnote in parity breaking, J. Théor. Nombres Bordeaux 30 (2018), no. 1, 219–225 (English, with English and French summaries). MR 3809717
- Michael Rubinstein, Computational methods and experiments in analytic number theory, Recent perspectives in random matrix theory and number theory, London Math. Soc. Lecture Note Ser., vol. 322, Cambridge Univ. Press, Cambridge, 2005, pp. 425–506. MR 2166470, DOI 10.1017/CBO9780511550492.015
- A. Sidi, Some properties of a generalization of the Richardson extrapolation process, J. Inst. Math. Appl. 24 (1979), no. 3, 327–346. MR 550478
- Avram Sidi, An algorithm for a special case of generalization of the Richardson extrapolation process, Numer. Math. 38 (1981/82), no. 3, 299–307. MR 654099, DOI 10.1007/BF01396434
- Avram Sidi, The numerical evaluation of very oscillatory infinite integrals by extrapolation, Math. Comp. 38 (1982), no. 158, 517–529. MR 645667, DOI 10.1090/S0025-5718-1982-0645667-5
- Avram Sidi, A user-friendly extrapolation method for oscillatory infinite integrals, Math. Comp. 51 (1988), no. 183, 249–266. MR 942153, DOI 10.1090/S0025-5718-1988-0942153-5
- Carl Ludwig Siegel, Contributions to the theory of the Dirichlet $L$-series and the Epstein zeta-functions, Ann. of Math. (2) 44 (1943), 143–172. MR 7760, DOI 10.2307/1968761
- Masaaki Sugihara, Optimality of the double exponential formula—functional analysis approach, Numer. Math. 75 (1997), no. 3, 379–395. MR 1427714, DOI 10.1007/s002110050244
- Burhan Sadiq and Divakar Viswanath, Finite difference weights, spectral differentiation, and superconvergence, Math. Comp. 83 (2014), no. 289, 2403–2427. MR 3223337, DOI 10.1090/S0025-5718-2014-02798-1
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- P. R. Taylor, On the Riemann zeta function, Quart. J. Math. Oxford Ser. 16 (1945), 1–21. MR 12626, DOI 10.1093/qmath/os-16.1.1
- I. J. Thompson and A. R. Barnett, Coulomb and Bessel functions of complex arguments and order, J. Comput. Phys. 64 (1986), no. 2, 490–509. MR 845195, DOI 10.1016/0021-9991(86)90046-X
- Joachim von zur Gathen and Jürgen Gerhard, Modern computer algebra, Cambridge University Press, New York, 1999. MR 1689167
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110