Arborified multiple zeta values. (English) Zbl 1459.11172
Burgos Gil, José Ignacio (ed.) et al., Periods in quantum field theory and arithmetic. Based on the presentations at the research trimester on multiple zeta values, multiple polylogarithms, and quantum field theory, ICMAT 2014, Madrid, Spain, September 15–19, 2014. Cham: Springer. Springer Proc. Math. Stat. 314, 469-481 (2020).
The paper deals with structures of multiple zeta values over the alphabet \(\{0,1\}\) (the simplest case possible), i.e., the long-known quasi-shuffle and stuffle relations and a series of more special relations. The Hopf-algebra structures tight up with the multiple zeta values of this kind is studied. Relations which can be derived from
J. Ecalle’s arborification [Ann. Inst. Fourier 42, No. 1–2, 73–164 (1992; Zbl 0940.32013)] are discussed. It remains open which particularly special new relations for multiple zeta values can be obtained in this way. A lot of essential literature, in particular after 2008 but also earlier, is unfortunately not mentioned. Here we refer the interested reader to the web-site https://www.usna.edu/Users/math/meh/biblio.html.
For the entire collection see [Zbl 1446.81002].
For the entire collection see [Zbl 1446.81002].
Reviewer: Johannes Blümlein (Zeuthen)
MSC:
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
| 16T05 | Hopf algebras and their applications |
| 81T17 | Renormalization group methods applied to problems in quantum field theory |
Citations:
Zbl 0940.32013Online Encyclopedia of Integer Sequences:
Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).
References:
| [1] | Brown, F.: On the decomposition of motivic multiple zeta values. Galois-Teichmüller theory and arithmetic geometry. Adv. Stud. Pure Math. 63, 31-58 (2012) · Zbl 1321.11087 |
| [2] | Brown, F.: Mixed Tate motives over \({\mathbb{Z}} \). Ann. Math. 175(1), 949-976 (2012) · Zbl 1278.19008 |
| [3] | Brown, F.: Depth-graded motivic multiple zeta values, preprint, arXiv:1301.3053 (2013) |
| [4] | Butcher, J.C.: An algebraic theory of integration methods. Math. Comp. 26, 79-106 (1972) · Zbl 0258.65070 |
| [5] | Cartier, P.: Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Séminaire Bourbaki No 885. Astérisque 282, 137-173 (2002) · Zbl 1085.11042 |
| [6] | Connes, A., Kreimer, D.: Hopf Algebras, Renormalization and Noncommutative Geometry. Comm. Math. Phys. 199, 203-242 (1998) · Zbl 0932.16038 |
| [7] | Deligne, P., Goncharov, A.: Groupes fondamentaux motiviques de Tate mixtes. Ann. Sci. Ec. Norm. Sup. (4) 38(1), 1-56 (2005) · Zbl 1084.14024 |
| [8] | Dür, A.: Möbius functions, incidence algebras and power series representations, Lect. Notes math. 1202, Springer (1986) · Zbl 0592.05006 |
| [9] | Ecalle, J.: Les fonctions résurgentes Vol. 2, Publications Mathématiques d’Orsay (1981). Available at http://portail.mathdoc.fr/PMO/feuilleter.php?id=PMO_1981 |
| [10] | Ecalle, J.: Singularités non abordables par la géométrie. Ann. Inst. Fourier 42(1-2), 73-164 (1992) · Zbl 0940.32013 |
| [11] | Ecalle, J., Vallet, B.: The arborification-coarborification transform: analytic, combinatorial, and algebraic aspects. Ann. Fac. Sci. Toulouse XIII(4), 575-657 (2004) · Zbl 1078.37016 |
| [12] | Eie, M., Liaw, W., Ong, Y.L.: A restricted sum formula for multiple zeta values. J. Number Theory 129, 908-921 (2009) · Zbl 1183.11053 |
| [13] | Fauvet, F., Menous, F.: Ecalle’s arborification-coarborification transforms and the Connes-Kreimer Hopf algebra. Ann. Sci. Ec. Norm. 50(1), 39-83 (2017) · Zbl 1371.16043 |
| [14] | Foissy, L.: Les algèbres de Hopf des arbres enracinés décorés I,II. Bull. Sci. Math. 126, 193-239, 249-288 (2002) · Zbl 1013.16027 |
| [15] | Hoffman, M.E.: Multiple harmonic series. Pacific J. Math. 152, 275-290 (1992) · Zbl 0763.11037 |
| [16] | Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Combin. 11, 49-68 (2000) · Zbl 0959.16021 |
| [17] | Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Comp. Math. 142(2), 307-338 (2004) · Zbl 1186.11053 |
| [18] | Kaneko, M., Yamamoto, S.: A new integral-series identity of multiple zeta values and regularizations. Selecta Math. 24(3), 2499-2521 (2018) · Zbl 1435.11114 |
| [19] | Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303-334 (1998) · Zbl 1041.81087 |
| [20] | Murua, A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Computational Math. 6, 387-426 (2006) · Zbl 1116.17004 |
| [21] | Racinet, G.: Doubles mélanges des polylogarithmes multiples aux racines de l’unité. Publ. Math. IHES 95, 185-231 (2002) · Zbl 1050.11066 |
| [22] | Schmitt, W.: Incidence Hopf algebras. J. Pure Appl. Algebra 96, 299-330 (1994) · Zbl 0808.05101 |
| [23] | Tanaka, T.: Restricted sum formula and derivation relation for multiple zeta values. arXiv:1303.0398 (2014) |
| [24] | Terasoma, T.: Mixed Tate motives and multiple zeta values. Invent. Math. 149(2), 339-369 (2002) · Zbl 1042.11043 |
| [25] | Yamamoto, S.: Multiple zeta-star values and multiple integrals, preprint. arXiv:1405.6499 (2014) · Zbl 1421.11069 |
| [26] | Zagier, D. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
