Symmetric Tornheim double zeta functions


Let \(s,t,u \in {{\mathbb {C}}}\) and T(stu) be the Tornheim double zeta function. In this paper, we investigate some properties of symmetric Tornheim double zeta functions which can be regarded as a desingularization of the Tornheim double zeta function. As a corollary, we give explicit evaluation formulas or rapidly convergent series representations for T(stu) in terms of series of the gamma function and the Riemann zeta function.

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  1. 1.

    For example, on \(s+t=1\), the function \((2^{s+t}-2)\zeta (s+t)\) has a possible (not true) singularity and the function \((2^{s+t}-2) \zeta ^2(s+t)\) has a true (not possible) singularity. In general, it is not easy to determine whether singularities of multivariable (zeta) functions are true or not.

  2. 2.

    they need some assumptions for \(x,y,z \in {{\mathbb {R}}}\) since T(stu) has the true singularities given by (1.2)

  3. 3.

    To state the main theorems simpler, we use this notation.


  1. 1.

    Apostol, T.M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York (1976)

    Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

    Borwein, J.M., Dilcher, K.: Derivatives and fast evaluation of the Tornheim zeta function. Ramanujan J. 45(2), 413–432 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

    Furusho, H., Komori, Y., Matsumoto, K., Tsumura, H.: Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type and evaluation of \(p\)-adic multiple \(L\)-functions at arbitrary integers. Various aspects of multiple zeta values, pp. 27–66, RIMS Kokyuroku Bessatsu, B68, Res. Inst. Math. Sci. (RIMS), Kyoto, (2017)

  6. 6.

    Laurinčikas, A., Garunkštis, R.: The Lerch zeta-function. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  7. 7.

    Matsumoto, K.: On the analytic continuation of various multiple zeta-functions in: Number Theory for the Millennium II, In: Proc. of the Millennial Conference on Number Theory, M. A. Bennett et. al. (eds.), A. K. Peters, pp. 417–440 (2002)

  8. 8.

    Matsumoto, K., Nakamura, T., Ochiai, H., Tsumura, H.: On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions. Acta Arith. 132(2), 99–125 (2008)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Nakamura, T.: A functional relation for the Tornheim double zeta function. Acta Arithmetica 125(3), 257–263 (2006)

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 11.

    Srivastava, H.M., Choi, J.: Zeta and q-Zeta functions and associated series and integrals. Elsevier Inc, Amsterdam (2012)

    Google Scholar 

  12. 12.

    Tsumura, H.: On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function. Math. Proc. Camb. Phil. Soc. 142, 395–405 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Whittaker, E.T., Watson, G.N.: A course of modern analysis. Reprint of the fourth: edition, p. 1996. Cambridge University Press, Cambridge, Cambridge Mathematical Library (1927)

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The author is partially supported by JSPS grant 16K05077. The author would like to thank Doctor Nobuo Sato who made formulas in Corollary 1.1 much simpler. The author would like to thank the referee also for a careful reading of the manuscript and valuable remarks and comments which have helped to get more higher quality of the paper.

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Correspondence to Takashi Nakamura.

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Nakamura, T. Symmetric Tornheim double zeta functions. Abh. Math. Semin. Univ. Hambg. (2021).

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  • Tornheim double zeta function
  • Desingularization
  • Rapidly convergent series representation

Mathematics Subject Classification

  • Primary 11M32