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Derivatives and fast evaluation of the Tornheim zeta function

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Abstract

We study analytic properties of the Tornheim zeta function \({\mathcal W}(r,s,t)\), which is also named after Mordell and Witten. In particular, we evaluate the function \({\mathcal W}(s,s,\tau s)\) (\(\tau >0\)) at \(s=0\) and, as our main result, find the derivative of this function at \(s=0\). Our principal tool is an identity due to Crandall that involves a free parameter and provides an analytic continuation. Furthermore, we derive special values of a permutation sum. Throughout this paper, we show by way of examples that Crandall’s identity can be used for efficient and high-precision evaluations of the Tornheim zeta function.

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References

  1. Agoh, T., Dilcher, K.: Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381(1), 10–16 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC (1964)

    MATH  Google Scholar 

  3. Bailey, D.H., Borwein, J.M.: Computation and theory of Mordell–Tornheim–Witten sums II. J. Approx. Theory 197, 115–140 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bailey, D.H., Borwein, J.M.: Computation and structure of character polylogarithms with applications to character Mordell–Tornheim–Witten sums. Math. Comput. 85(297), 295–324 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bailey, D.H., Borwein, J.M.: Computation and experimental evaluation of Mordell–Tornheim–Witten sum derivatives. Exp. Math. (to appear)

  6. Borwein, J.M.: Hilbert’s inequality and Witten’s zeta-function. Am. Math. Monthly 115(2), 125–137 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Borwein, J.M., Bailey, D.H.: Mathematics by Experiment. Plausible Reasoning in the 21st Century, 2nd edn. A K Peters Ltd., Wellesley (2008)

    MATH  Google Scholar 

  8. Borwein, J.M., Dilcher, K.: Analytic continuation and derivatives of character Mordell–Tornheim–Witten sums (in preparation)

  9. Borwein, J.M., Dilcher, K., Tomkins, H.: The behaviour at the origin of multiple Witten zeta functions (in preparation)

  10. Borwein, J.M., Zucker, I.J., Boersma, J.: The evaluation of character Euler double sums. Ramanujan J. 15, 377–405 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carlitz, L.: Note on the integral of the product of several Bernoulli polynomials. J. Lond. Math. Soc. 34, 361–363 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  12. Crandall, R.E.: Unified algorithms for polylogarithm, L-series, and zeta variants. In: Algorithmic Reflections: Selected Works. PSI Press, Portland (2012)

  13. Crandall, R.E., Buhler, J.P.: On the evaluation of Euler sums. Exp. Math. 3(4), 275–285 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dilcher, K.: A nonlinear identity of Bernoulli polynomials (2016). (Preprint)

  15. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, Vol. I. Based, in part, on notes left by Harry Bateman. McGraw-Hill, New York (1953)

  16. Espinosa, O., Moll, V.H.: The evaluation of Tornheim double sums. I. J. Number Theory 116(1), 200–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007)

    MATH  Google Scholar 

  18. Kurokawa, N., Ochiai, H.: Zeros of Witten zeta functions and absolute limit. Kodai Math. J. 36(3), 440–454 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Matsumoto, K.: On the analytic continuation of various multiple zeta-functions. In: Number Theory for the Millennium, II (Urbana: pp. 417–440, 2000). A K Peters, Natick (2002)

  20. Mordell, L.J.: On the evaluation of some multiple series. J. Lond. Math. Soc. 33, 368–371 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  21. Olver, F.W.J. (ed.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)

    MATH  Google Scholar 

  22. Onodera, K.: Mordell–Tornheim multiple zeta values at non-positive integers. Ramanujan J. 32(2), 221–226 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Onodera, K.: A functional relation for Tornheim’s double zeta functions. Acta Arith. 162(4), 337–354 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Romik, D.: On the number of \(n\)-dimensional representations of \(SU(3)\), the Bernoulli numbers, and the Witten zeta function (2015)

  25. Subbarao, M.V., Sitaramachandra, R.: Rao, On some infinite series of L. J. Mordell and their analogues. Pac. J. Math. 119(1), 245–255 (1985)

    Article  Google Scholar 

  26. Tomkins, H.: An exploration of multiple zeta functions. Honours Thesis, Dalhousie University (2016)

  27. Tornheim, L.: Harmonic double series. Am. J. Math. 72, 303–314 (1950)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We dedicate this paper to the memory of Richard E. Crandall (1947–2012), without whose insights and inspiration this work would not have been possible. We also thank Hayley Tomkins for helpful comments which led to an improved exposition. Finally, we thank one of the referees for bringing the papers [18], [22], and [23] to our attention.

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Authors and Affiliations

  1. CARMA, University of Newcastle, Callaghan, NSW, 2308, Australia

    Jonathan M. Borwein

  2. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 4R2, Canada

    Karl Dilcher

Authors
  1. Jonathan M. Borwein
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  2. Karl Dilcher
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Correspondence to Karl Dilcher.

Additional information

J. M. Borwein passed away on Aug. 2, 2016, while this paper was under review. Research supported in part by the Natural Sciences and Engineering Research Council of Canada.

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Borwein, J.M., Dilcher, K. Derivatives and fast evaluation of the Tornheim zeta function. Ramanujan J 45, 413–432 (2018). https://doi.org/10.1007/s11139-017-9890-9

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  • DOI: https://doi.org/10.1007/s11139-017-9890-9

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