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On Hoffman’s \(t\)-values of maximal height and generators of multiple zeta values. (English) Zbl 1504.11091

In conjunction with multiple zeta values (MZV), the multiple \(t\)-value (MtV) is defined by \[ t(k_1,k_2,\dots,k_p)=\sum_{\substack{0<m_1<m_2<\dots <m_p\\ m_i:\text{ odd}}} \frac{1}{m_1^{k_1}m_2^{k_2}\cdots m_p^{k_p}}. \] It is well known that any multiple \(t\)-value can be written as a \(\mathbb{Q}\)-linear combination of Euler sums. In the paper under review, the author first shows that when all \(k_i\) are greater than \(1\), \(t(k_1,\dots, k_p)\) is a \(\mathbb{Q}\)-linear combination of multiple zeta values (MZV). Conversely, it is shown that every multiple zeta value is a \(\mathbb{Q}\)-linear combination of multiple \(t\)-value \(t(k_1,\dots, k_p)\) with \(k_i\in \lbrace 2, 3\rbrace\). Finally, the author proves some results on multiple \(t\)-values, as conjectured by M. Kaneko and H. Tsumura in their paper on multiple zeta values of level two [Tsukuba J. Math. 44, No. 2, 213–234 (2020; Zbl 1469.11327)].

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values

Citations:

Zbl 1469.11327
Full Text: DOI

References:

[1] Brown, F., Mixed Tate motives over \({\mathbb{Z}} \), Ann. Math., 175, 1, 949-976 (2012) · Zbl 1278.19008 · doi:10.4007/annals.2012.175.2.10
[2] Brown, F.: Notes on motivic periods. arXiv:1512.06410v3 · Zbl 1390.14024
[3] Deligne, P., Goncharov, A.B.: Groupes fondamentaux motiviques de Tate mixte, dans Ann. Scient. Ecole. Norm. Sup., 4e serie, t. 38, pp. 1-56 (2005) · Zbl 1084.14024
[4] Glanois, C.: Periods of the motivic fundamental groupoid of \({\mathbb{P}}^1\backslash \{0,\mu_N,\infty \} \), PhD thesis. Université Pierre et Marie Curie (2016) · Zbl 1398.14011
[5] Glanois, C.: Unramified Euler sums and Hoffman \(\star\) basis (preprint). arXiv:1603.05178 · Zbl 1398.14011
[6] Kaneko, M., Tsumura, H.: Zeta functions connecting multiple zeta values and poly-Bernoulli numbers. Adv. Stud. Pure Math. (to appear) · Zbl 1476.11120
[7] Kaneko, M., Tsumura, H.: On a variant of multiple zeta values of level two. Tsukuba J. Math. (to appear) · Zbl 1469.11327
[8] Hoffman, ME, An odd variant of multiple zeta values, Commun. Number Theory Phys., 13, 3, 529-567 (2019) · Zbl 1447.11095
[9] Nielsen, N., Handbuch der Theorie der Gammafunktion (1906), Leipzig: B.G. Teubner, Leipzig · JFM 37.0450.01
[10] Terasoma, T., Mixed Tate motives and multiple zeta values, Invent. Math., 149, 339-369 (2002) · Zbl 1042.11043 · doi:10.1007/s002220200218
[11] Zagier, DB, Evaluation of the multiple zeta values \(\zeta (2,\dots ,2,3,2,\dots 2)\), Ann. Math., 175, 977-1000 (2012) · Zbl 1268.11121 · doi:10.4007/annals.2012.175.2.11
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