## Computation and theory of extended Mordell-Tornheim-Witten sums

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- 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

## Additional Information

**David H. Bailey**- Affiliation: Lawrence Berkeley National Laboratory, Berkeley, California 94720
- MR Author ID: 29355
- Email: DHBailey@lbl.gov
**Jonathan M. Borwein**- Affiliation: CARMA, University of Newcastle, Callaghan, NSW 2308, Australia and Distinguished Professor King Abdulaziz University, Jeddah
- Email: jonathan.borwein@newcastle.edu.au
**Richard E. Crandall**- Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202
- Email: crandall@reed.edu
- Received by editor(s): April 26, 2012
- Received by editor(s) in revised form: August 1, 2012
- Published electronically: January 23, 2014
- Additional Notes: Richard Crandall passed away on December 20, 2012

**Copyright Status:**LBNL authored documents are sponsored by the U.S. Department of Energy under Contract DE-AC02-05CH11231. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce these documents, or allow others to do so, for U.S. Government purposes. The documents may be freely distributed and used for noncommercial, scientific and educational purposes. - Journal: Math. Comp.
**83**(2014), 1795-1821 - MSC (2010): Primary 33B15, 33F05, 65D20, 65D30, 11M32
- DOI: https://doi.org/10.1090/S0025-5718-2014-02768-3
- MathSciNet review: 3194130