Computation and structure of character polylogarithms with applications to character Mordell–Tornheim–Witten sums
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- by D. H. Bailey and J. M. Borwein;
- Math. Comp. 85 (2016), 295-324
- DOI: https://doi.org/10.1090/mcom/2974
- Published electronically: June 3, 2015
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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
Bibliographic 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