## Computation and structure of character polylogarithms with applications to character Mordell–Tornheim–Witten sums

HTML articles powered by AMS MathViewer

- by D. H. Bailey and J. M. Borwein PDF
- Math. Comp.
**85**(2016), 295-324 Request permission

## Abstract:

This paper extends tools developed to study character polylogarithms. These objects are used to compute Mordell–Tornheim–Witten character sums and to explore their connections with multiple-zeta values (MZVs) and with their character analogues.## References

- Scott Ahlgren, Bruce C. Berndt, Ae Ja Yee, and Alexandru Zaharescu,
*Integrals of Eisenstein series and derivatives of $L$-functions*, Int. Math. Res. Not.**32**(2002), 1723–1738. MR**1916839**, DOI 10.1155/S107379280211110X - Tom M. Apostol,
*Dirichlet $L$-functions and character power sums*, J. Number Theory**2**(1970), 223–234. MR**258766**, DOI 10.1016/0022-314X(70)90022-3 - Tom M. Apostol,
*Euler’s $\phi$-function and separable Gauss sums*, Proc. Amer. Math. Soc.**24**(1970), 482–485. MR**257006**, DOI 10.1090/S0002-9939-1970-0257006-6 - Tom M. Apostol,
*Dirichlet $L$-functions and primitive characters*, Proc. Amer. Math. Soc.**31**(1972), 384–386. MR**285499**, DOI 10.1090/S0002-9939-1972-0285499-9 - 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., E-published February 2013. Available at http://link.springer.com/article/10.1007/s11139-012-9427-1. - D. H. Bailey and J. M. Borwein,
*Computation and theory of extended Mordell-Tornheim-Witten sums II*, Journal of Approximation Theory, Submitted. Available at http://carma.newcastle.edu.au/jon/MTW2.pdf. - D. H. Bailey and J. M. Borwein,
*Crandall’s computation of the incomplete Gamma function and the Hurwitz zeta function with applications to Dirichlet L-series*Submitted. Available at http://carma.newcastle.edu.au/jon/lerch.pdf. - David H. Bailey, Jonathan M. Borwein, and Richard E. Crandall,
*Computation and theory of extended Mordell-Tornheim-Witten sums*, Math. Comp.**83**(2014), no. 288, 1795–1821. MR**3194130**, DOI 10.1090/S0025-5718-2014-02768-3 - 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 - 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, David Bailey, and Roland Girgensohn,
*Experimentation in mathematics*, A K Peters, Ltd., Natick, MA, 2004. Computational paths to discovery. MR**2051473** - Jonathan M. Borwein and Peter B. Borwein,
*Pi and the AGM*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. A study in analytic number theory and computational complexity; A Wiley-Interscience Publication. MR**877728** - J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker,
*Lattice sums then and now*, Encyclopedia of Mathematics and its Applications, vol. 150, Cambridge University Press, Cambridge, 2013. With a foreword by Helaman Ferguson and Claire Ferguson. MR**3135109**, DOI 10.1017/CBO9781139626804 - 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 - J. M. Borwein, I. J. Zucker, and J. Boersma,
*The evaluation of character Euler double sums*, Ramanujan J.**15**(2008), no. 3, 377–405. MR**2390277**, DOI 10.1007/s11139-007-9083-z - 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** - R. Crandall,
*Unified algorithms for polylogarithm, L-series, and zeta variants*. In*Algorithmic Reflections: Selected Works*. PSIpress, 2012. Sadly, Crandall died in 2012, and PSIpress is now defunct; however, some copies of this collection are available from third-party booksellers. - Christopher Deninger,
*On the analogue of the formula of Chowla and Selberg for real quadratic fields*, J. Reine Angew. Math.**351**(1984), 171–191. MR**749681**, DOI 10.1515/crll.1984.351.171 - A. Erdélyi,
*Higher Transcendental Functions. Volumes 1, 2, 3*, Malabar, FL: Krieger. ISBN 0-486-44614-X. ISBN 0-486-44615-8. ISBN 0-486-44616-6, 1981. - 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 - F. Johansson,
*Rigorous high-precision computation of the Hurwitz zeta function and its derivatives*, 2013. Available at http://arxiv.org/abs/1309.2877. - L. Lewin,
*On the evaluation of log-sine integrals*, The Mathematical Gazette,**42**(1958), 125–128. - 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** - Takashi Nakamura,
*Double Lerch series and their functional relations*, Aequationes Math.**75**(2008), no. 3, 251–259. MR**2424133**, DOI 10.1007/s00010-007-2921-7 - Takashi Nakamura,
*Double Lerch value relations and functional relations for Witten zeta functions*, Tokyo J. Math.**31**(2008), no. 2, 551–574. MR**2477890**, DOI 10.3836/tjm/1233844070 - Takashi Nakamura,
*A simple proof of the functional relation for the Lerch type Tornheim double zeta function*, Tokyo J. Math.**35**(2012), no. 2, 333–337. MR**3058710**, DOI 10.3836/tjm/1358951322 - F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark,
*NIST Digital Handbook of Mathematical Functions*, 2012. - Leonard Tornheim,
*Harmonic double series*, Amer. J. Math.**72**(1950), 303–314. MR**34860**, DOI 10.2307/2372034 - Hirofumi Tsumura,
*On alternating analogues of Tornheim’s double series*, Proc. Amer. Math. Soc.**131**(2003), no. 12, 3633–3641. MR**1998168**, DOI 10.1090/S0002-9939-03-07186-7 - 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,
*Evaluation formulas for Tornheim’s type of alternating double series*, Math. Comp.**73**(2004), no. 245, 251–258. MR**2034120**, DOI 10.1090/S0025-5718-03-01572-2 - 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 - Hirofumi Tsumura,
*On alternating analogues of Tornheim’s double series. II*, Ramanujan J.**18**(2009), no. 1, 81–90. MR**2471618**, DOI 10.1007/s11139-007-9016-x - Jianqiang Zhao,
*A note on colored Tornheim’s double series*, Integers**10**(2010), A59, 879–882. MR**2797781**, DOI 10.1515/INTEG.2010.059 - Xia Zhou, Tianxin Cai, and David M. Bradley,
*Signed $q$-analogs of Tornheim’s double series*, Proc. Amer. Math. Soc.**136**(2008), no. 8, 2689–2698. MR**2399030**, DOI 10.1090/S0002-9939-08-09208-3

## Additional Information

**D. H. Bailey**- Affiliation: Lawrence Berkeley National Lab (retired), Berkeley, California 94720 – and – Department of Computer Science, University of California, Davis, Davis, California 95616
- MR Author ID: 29355
- Email: david@davidhbailey.com
**J. M. Borwein**- Affiliation: CARMA, University of Newcastle, NSW 2303, Australia
- Email: jon.borwein@gmail.com
- Received by editor(s): February 25, 2014
- Received by editor(s) in revised form: June 9, 2014, and June 18, 2014
- Published electronically: June 3, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp.
**85**(2016), 295-324 - MSC (2010): Primary 11L40, 33F05
- DOI: https://doi.org/10.1090/mcom/2974
- MathSciNet review: 3404451