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

2020/09/09 by Masataka Ono, Shin-ichiro Seki, Ono, Masataka +3 · 1 citation
Mathematics · #05C05 #11M32 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2009.04112

openalex publication_date 2020/09/09 · openalex created_date 2020/09/14 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Multiple polylogarithms and mixed Tate motives
• Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values
• On variants of symmetric multiple zeta-star values and the cyclic sum formula

Cited by

• A Lie algebra associated with adjoint multiple zeta values

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