Truncated t-adic symmetric multiple zeta values and double shuffle relations


We study a refinement of the symmetric multiple zeta value, called the t-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the t-adic symmetric multiple zeta values, thus we introduce two kinds of truncations correspondingly. Then we show that our truncations tend to the corresponding t-adic symmetric multiple zeta values, and satisfy the harmonic and shuffle relations, respectively. This gives a new proof of the double shuffle relations for t-adic symmetric multiple zeta values, first proved by Jarossay. In order to prove the shuffle relation, we develop the theory of truncated t-adic symmetric multiple zeta values associated with 2-colored rooted trees. Finally, we discuss a refinement of Kaneko–Zagier’s conjecture and the t-adic symmetric multiple zeta values of Mordell–Tornheim type.

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The authors would like to thank Dr. Minoru Hirose for communicating his idea on the definition of \(\widehat{\mathcal {S}}\)-MZV. They also would like to thank Dr. Yuta Suzuki for helpful comments and useful discussion on Proposition 4.4. They would like to express their sincere gratitude to Prof. Koji Tasaka for informing us about Jarossay’s work on the \(\Lambda \)-adjoint multiple zeta values, and to Dr. David Jarossay for explaining in detail the relationship of his work with ours.

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Correspondence to Masataka Ono.

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This research was supported in part by JSPS KAKENHI Grant Numbers 26247004, 16J01758, JP16H06336, 18J00151, 18K03221, 18H05233.

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Ono, M., Seki, Si. & Yamamoto, S. Truncated t-adic symmetric multiple zeta values and double shuffle relations. Res. number theory 7, 15 (2021).

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  • t-adic symmetric multiple zeta values
  • Double shuffle relation
  • Kaneko–Zagier’s conjecture
  • Multiple zeta values of Mordell–Tornheim type

Mathematics Subject Classification

  • 11M32
  • 05C05