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A new integral–series identity of multiple zeta values and regularizations

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Abstract

We present a new “integral \(=\) series” type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear relations of multiple zeta values over \(\mathbb {Q}\). We also establish the regularization theorem for multiple zeta-star values, which too is equivalent to our new identity. A connection to Kawashima’s relation is discussed as well.

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Acknowledgements

The authors are very grateful to the referee for his/her careful reading and valuable comments. This work was supported in part by JSPS KAKENHI Grant Nos. JP24224001, JP23340010, JP26247004, JP16H06336, as well as JSPS Joint Research Project with CNRS “Zeta functions of several variables and applications,” JSPS Core-to-Core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry” and the KiPAS program 2013–2018 of the Faculty of Science and Technology at Keio University.

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Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Masanobu Kaneko

  2. Keio Institute of Pure and Applied Sciences (KiPAS), Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Shuji Yamamoto

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  1. Masanobu Kaneko
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  2. Shuji Yamamoto
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Correspondence to Shuji Yamamoto.

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Kaneko, M., Yamamoto, S. A new integral–series identity of multiple zeta values and regularizations. Sel. Math. New Ser. 24, 2499–2521 (2018). https://doi.org/10.1007/s00029-018-0400-8

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  • DOI: https://doi.org/10.1007/s00029-018-0400-8

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