Symmetric Tornheim double zeta functions

Abstract

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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Notes

  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.

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Acknowledgements

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). https://doi.org/10.1007/s12188-021-00232-4

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Keywords

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

Mathematics Subject Classification

  • Primary 11M32