Computation and theory of extended Mordell-Tornheim-Witten sums

Bailey, David; Borwein, Jonathan; Crandall, Richard

doi:10.1090/S0025-5718-2014-02768-3

Computation and theory of extended Mordell-Tornheim-Witten sums
HTML articles powered by AMS MathViewer

by David H. Bailey, Jonathan M. Borwein and Richard E. Crandall PDF

Math. Comp. 83 (2014), 1795-1821 Request permission

Abstract:

We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zeta-function values along with their derivatives, and explore connections with multiple-zeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. Our original motivation was to represent unresolved constructs such as Eulerian log-gamma integrals. We are able to resolve all such integrals in terms of an MTW basis. We also present, for a substantial subset of MTW values, explicit closed-form expressions. In the process, we significantly extend methods for high-precision numerical computation of polylogarithms and their derivatives with respect to order.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Math. Comp. 44 (1985), no. 169, 223–232. MR 771044, DOI 10.1090/S0025-5718-1985-0771044-5
D. H. Bailey, R. Barrio, and J. M. Borwein, High-precision computation: mathematical physics and dynamics, Appl. Math. Comput. 218 (2012), no. 20, 10106–10121. MR 2921767, DOI 10.1016/j.amc.2012.03.087
D. Bailey, D. Borwein, and J. Borwein, On Eulerian log–Gamma integrals and Tornheim–Witten zeta functions, Ramanujan J., Feb. 27, 2013, DOI 10.1007/s11139-012-9427-1.
David H. Bailey, Jonathan M. Borwein, and Richard E. Crandall, On the Khintchine constant, Math. Comp. 66 (1997), no. 217, 417–431. MR 1377659, DOI 10.1090/S0025-5718-97-00800-4
David H. Bailey and David J. Broadhurst, Parallel integer relation detection: techniques and applications, Math. Comp. 70 (2001), no. 236, 1719–1736. MR 1836930, DOI 10.1090/S0025-5718-00-01278-3
D. H. Bailey, Y. Hida, X. S. Li, and B. Thompson, ARPREC: An arbitrary precision computation package, 2002.
David H. Bailey, Karthik Jeyabalan, and Xiaoye S. Li, A comparison of three high-precision quadrature schemes, Experiment. Math. 14 (2005), no. 3, 317–329. MR 2172710, DOI 10.1080/10586458.2005.10128931
David Borwein, Jonathan M. Borwein, and Roland Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 277–294. MR 1335874, DOI 10.1017/S0013091500019088
David Borwein, Jonathan M. Borwein, Armin Straub, and James Wan, Log-sine evaluations of Mahler measures, II, Integers 12 (2012), no. 6, 1179–1212. MR 3011556, DOI 10.1515/integers-2012-0035
Jonathan M. Borwein, Hilbert’s inequality and Witten’s zeta-function, Amer. Math. Monthly 115 (2008), no. 2, 125–137. MR 2384265, DOI 10.1080/00029890.2008.11920505
Jonathan Borwein and David Bailey, Mathematics by experiment, 2nd ed., A K Peters, Ltd., Wellesley, MA, 2008. Plausible reasoning in the 21st Century. MR 2473161, DOI 10.1201/b10704
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, and Petr Lisoněk, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), no. 3, 907–941. MR 1709772, DOI 10.1090/S0002-9947-00-02616-7
Jonathan M. Borwein, David M. Bradley, and Richard E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math. 121 (2000), no. 1-2, 247–296. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780051, DOI 10.1016/S0377-0427(00)00336-8
Jonathan M. Borwein and Armin Straub, Special values of generalized log-sine integrals, ISSAC 2011—Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2011, pp. 43–50. MR 2895193, DOI 10.1145/1993886.1993899
P. Borwein, An efficient algorithm for the Riemann zeta function, Constructive, experimental, and nonlinear analysis (Limoges, 1999) CRC Math. Model. Ser., vol. 27, CRC, Boca Raton, FL, 2000, pp. 29–34. MR 1777614
David M. Bradley and Xia Zhou, On Mordell-Tornheim sums and multiple zeta values, Ann. Sci. Math. Québec 34 (2010), no. 1, 15–23 (English, with English and French summaries). MR 2744193
Bejoy K. Choudhury, The Riemann zeta-function and its derivatives, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1940, 477–499. MR 1356175, DOI 10.1098/rspa.1995.0096
Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence acceleration of alternating series, Experiment. Math. 9 (2000), no. 1, 3–12. MR 1758796, DOI 10.1080/10586458.2000.10504632
R. Crandall,Unified algorithms for polylogarithm, L-series, and zeta variants, 2012. Available at http://www.perfscipress.com/papers/UniversalTOC25.pdf.
Olivier Espinosa and Victor H. Moll, The evaluation of Tornheim double sums. I, J. Number Theory 116 (2006), no. 1, 200–229. MR 2197867, DOI 10.1016/j.jnt.2005.04.008
Olivier Espinosa and Victor H. Moll, The evaluation of Tornheim double sums. II, Ramanujan J. 22 (2010), no. 1, 55–99. MR 2610609, DOI 10.1007/s11139-009-9181-1
M. Kalmykov, About higher order $\epsilon$-expansion of some massive two- and three-loop master-integrals, Nuclear Physics B, 718:276–292, July 2005.
Yasushi Komori, An integral representation of the Mordell-Tornheim double zeta function and its values at non-positive integers, Ramanujan J. 17 (2008), no. 2, 163–183. MR 2452649, DOI 10.1007/s11139-008-9130-4
L. Lewin, On the evaluation of log-sine integrals, The Mathematical Gazette, 42:125–128, 1958.
Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
Kohji Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, vol. 360, Univ. Bonn, Bonn, 2003, pp. 17. MR 2075634
N. Nielsen, Handbuch der theorie der Gammafunction, Druck und Verlag von B.G. Teubner, Leipzig, 1906.
Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Digital Handbook of Mathematical Functions, 2012.
Kohji Matsumoto, Analytic properties of multiple zeta-functions in several variables, Number theory, Dev. Math., vol. 15, Springer, New York, 2006, pp. 153–173. MR 2213834, DOI 10.1007/0-387-30829-6_{1}1
Kazuhiro Onodera, Generalized log sine integrals and the Mordell-Tornheim zeta values, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1463–1485. MR 2737273, DOI 10.1090/S0002-9947-2010-05176-1
Leonard Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303–314. MR 34860, DOI 10.2307/2372034
Hirofumi Tsumura, Combinatorial relations for Euler-Zagier sums, Acta Arith. 111 (2004), no. 1, 27–42. MR 2038060, DOI 10.4064/aa111-1-3
Hirofumi Tsumura, On Mordell-Tornheim zeta values, Proc. Amer. Math. Soc. 133 (2005), no. 8, 2387–2393. MR 2138881, DOI 10.1090/S0002-9939-05-08132-3