Abstract
The Kaneko–Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous works with H. Bachmann and Y. Takeyama, we proved that the finite and symmetric multiple zeta values are obtained as an ‘algebraic’ and ‘analytic’ limit at \(q\rightarrow 1\) of certain multiple harmonic q-sums, and studied their relations in order to give partial evidence of the Kaneko–Zagier conjecture. In this paper, we start with multiple harmonic q-sums of level N, which are q-analogues of the truncated colored multiple zeta values. We introduce our finite and symmetric colored multiple zeta values as an algebraic and analytic limit of the multiple harmonic q-sums of level N and discuss a higher level (or a cyclotomic) analogue of the Kaneko–Zagier conjecture.
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References
Arakawa, T., Kaneko, M.: On multiple L-values. J. Math. Soc. Jpn. 56, 967–991 (2004)
Bachmann, H., Takeyama, Y., Tasaka, K.: Cyclotomic analogues of finite multiple zeta values. Compos. Math. 154(12), 2701–2721 (2018)
Bachmann, H., Takeyama, Y., Tasaka, K.: Special values of finite multiple harmonic q-series at roots of unity. Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA), vol. 2, IRMA Lectures in Mathematics and Theoretical Physics 32 (EMS), pp. 1–18 (2020)
Brown, F.: Mixed Tate motives over \({\mathbb{Z}}\). Ann. Math. 175(2), 949–976 (2012)
Chen, K.T.: Iterated path integrals. Bull. Am. Math. Soc. 83, 831–879 (1977)
Deligne, P., Goncharov, A.B.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. 38, 1–56 (2005)
Deligne, P.: Le groupe fondamental unipotent motivique de \(\mathbb{G}_m\backslash \mu _N\) pour \(N = 2, 3, 4, 6\) or \(8\). Publ. Math. Inst. Hautes Études Sci. 112(1), 101–141 (2010)
Glanois, C.: Motivic unipotent fundamental groupoid of \(\mathbb{G}_m \backslash \mu _N\) for \(N = 2, 3, 4, 6, 8\) and Galois descents. J. Number Theory 160, 334–384 (2016)
Goncharov, A.B.: Multiple polylogarithms, cyclotomy, and modular complexes. Math. Res. Lett. 5, 497–516 (1998)
Hirose, M.: Double shuffle relations for refined symmetric multiple zeta values. Doc. Math. 25, 365–380 (2020)
Hoffman, M.E.: The algebra of multiple harmonic series. J. Algebra 194(2), 477–495 (1997)
Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)
Jarossay, D.: Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values (preprint (v4))
Kaneko, M.: An introduction to classical and finite multiple zeta values. Publ. Math. Besancon 1, 103–129 (2019)
Kaneko, M., Zagier, D.: Finite multiple zeta values (in preparation)
The PARI Group, PARI/GP version 2.11.0, Univ. Bordeaux (2018). http://pari.math.u-bordeaux.fr/
Racinet, G.: Doubles mélanges des polylogarithmes multiples aux racines de lúnité. Publ. Math. IHES 95, 185–231 (2002)
Singer, J., Zhao, J.: Finite and symmetrized colored multiple zeta values. Finite Fields Appl. 65, 101676 (2020)
Takeyama, Y.: Derivations on the algebra of multiple harmonic \(q\)-series and their applications. Ramanujan J. 54, 41–65 (2020)
Tasaka, K.: Congruence model of finite and symmetric multiple zeta values (in preparation)
Washington, L.: Introduction to Cyclotomic Fields, GTM 83. Springer, New York (1997)
Yasuda, S.: Finite real multiple zeta values generate the whole space \(\cal{Z}\). Int. J. Number Theory 12(3), 787–812 (2016)
Zhao, J.: Standard relations of multiple polylogarithm values at roots of unity. Doc. Math. 15, 1–34 (2010)
Zhao, J.: Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values. World Scientific, Singapore (2016)
Acknowledgements
The author is grateful to Henrik Bachmann and Yoshihiro Takeyama for very valuable discussions. The author is also very grateful to Jianqiang Zhao for helpful comment on the linear shuffle relation. This work was partially supported by JSPS KAKENHI Grant Numbers 18K13393.
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Tasaka, K. Finite and symmetric colored multiple zeta values and multiple harmonic q-series at roots of unity. Sel. Math. New Ser. 27, 21 (2021). https://doi.org/10.1007/s00029-021-00636-3
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DOI: https://doi.org/10.1007/s00029-021-00636-3