Bachmann–Kühn's brackets and multiple zeta values at level N

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• 作者：Haiping Yuan ; Jianqiang Zhao
• 刊名：manuscripta mathematica
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：150
• 期：1-2
• 页码：177-210
• 全文大小：673 KB
• 参考文献：1.Bachmann, H., Kühn, U.: The algebra of multiple divisor functions and applications to multiple zeta values, preprint at arXiv:​ 1309.​3920
  2.Bachmann, H.: Multiple zeta-Werte und de Verbindung zu Modulformen durch Multiple Eisensteinreihen. Master thesis, Hamburg Univesity (2012)
  3.Blümlein J., Broadhurst D.J., Vermaseren J.A.M.: The multiple zeta value data mine. Comput. Phys. Commun. 181, 582–625 (2010)MathSciNet CrossRef MATH
  4.Borwein J.M., Bradley D.M.: Thirty-two Goldbach variations. Int. J. Number Theory 2(1), 65–103 (2006)MathSciNet CrossRef MATH
  5.Broadhurst, D.J.: Conjectured enumeration of irreducible multiple zeta values, from knots and Feynman diagrams, preprint at arXiv:​ hep-th/​9612012
  6.Broadhurst D.J.: Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity. Eur. Phys. J. C (Fields) 8, 311–333 (1999)MathSciNet
  7.Deligne P., Goncharov A.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. Ecole Norm. S. 38(1), 1–56 (2005)MathSciNet
  8.Eulerian polynomials. http://​oeis.​org/​wiki/​Eulerian_​polynomials . Accessed on 10 Feb 2014
  9.Foata, D.: Eulerian polynomials: from Euler’s time to the present. In: Alladi, K., Klauder, J.R., Calyampudi, R.R. (eds.) The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, pp. 253–273. Springer, New York (2010)
  10.Gangl H., Kaneko M., Zagier D.: Double zeta values and modular forms. In: Böcherer, S., Ibukiyama, T., Kaneko, M., Sato, F. (eds) Automorphic Forms and Zeta Functions, pp. 71–106. World Scientific, Hackensack (2006)CrossRef
  11.Hoffman M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992)MathSciNet CrossRef MATH
  12.Hoffman M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997)MathSciNet CrossRef MATH
  13.Hoffman M., Ihara, K.: Quasi-shuffle products revisited, Max-Planck-Institut für Mathematik Preprint Series 16 (2012)
  14.Ihara K., Kaneko M., Zagier D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)MathSciNet CrossRef MATH
  15.Kaneko M., Tasaka K.: Double zeta values, double Eisenstein series, and modular forms of level 2. Math. Ann. 357(3), 1091–1118 (2013)MathSciNet CrossRef MATH
  16.Nakamura T., Tasaka K.: Remarks on double zeta values of level 2. J. Number Theory 133, 48–54 (2013)MathSciNet CrossRef MATH
  17.Pupyrev, Yu.: Linear and algebraic independence of q-zeta values. Math. Notes 78(4), 563–568 (2005). Translated from Mate. Zametki 78(4), 608–613 (2005)
  18.Racinet G.: Doubles mélanges des polylogarithmes multiples aux racines de l’unité. Publ. Math. IHES 95, 185–231 (2002)MathSciNet CrossRef
  19.Reutenauer, C.: Free Lie Algebra, London Mathematical Society Monographs, New Series, No. 7. Oxford University Press, Oxford (1993)
  20.Yuan, H., Zhao, J.: Double shuffle relations of double zeta values and double eisenstein series at level N. J. Lond. Math. Soc. doi:​10.​1112/​jlms/​jdv042
  21.Zagier, D.: Values of zeta functions and their applications. In: Joseph, A., et al. (eds.) First European Congress of Mathematics (Paris, 1992), vol. II, pp. 497–512. Birkhäuser, Basel (1994)
  22.Zhao J.: Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J. 14(2), 189–221 (2007)MathSciNet CrossRef MATH
  23.Zhao J.: Analytic continuation of multiple polylogarithms. Anal. Math. 33, 301–323 (2007)MathSciNet CrossRef MATH
  24.Zhao J.: Multiple polylogarithm values at roots of unity. C. R. Acad. Sci. Paris Ser. I. 346, 1029–1032 (2008)MathSciNet CrossRef MATH
  25.Zhao, J.: Double shuffle relations of euler sums, unpublished manuscript. arXiv:​0705.​2267 . [26] is the abridged published version
  26.Zhao J.: On a conjecture of Borwein, Bradley and Broadhurst. J. Reine Angew. Math. 639, 223–233 (2010)MathSciNet MATH
  27.Zhao J.: Standard relations of multiple polylogarithm values at roots of unity. Doc. Math. 15, 1–34 (2010)MathSciNet MATH
  
• 作者单位：Haiping Yuan (1)
  Jianqiang Zhao (2)
  
  1. Department of Physical Sciences, York College of Pennsylvania, York, PA, 17403, USA
  2. Department of Mathematics, Eckerd College, St. Petersburg, FL, 33711, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebraic Geometry
  Topological Groups and Lie Groups
  Geometry
  Number Theory
  Calculus of Variations and Optimal Control
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1785

文摘

Multiple zeta values (MZVs) are generalizations of Riemann zeta values at positive integers to multiple variable setting. These values can be further generalized to level N multiple polylog values by evaluating multiple polylogs at Nth roots of unity. In this paper, we consider another level N generalization by restricting the indices in the iterated sums defining MZVs to congruence classes modulo N, which we call the MZVs at level N. The goals of this paper are twofold. First, we shall lay down the theoretical foundations of these values such as their regularizations and double shuffle relations. Second, we will generalize the bracket functions related to multiple divisor sums defined by Bachmann and Kühn to arbitrary level N and study their relations to MZVs at level N. The brackets are all q-series and similar to MZVs, they have both weight and depth filtrations. But unlike that of MZVs, the product of brackets usually has mixed weights; however, after projecting to the highest weight we can obtain an algebra homomorphism from brackets to MZVs. Moreover, the image of the derivation \({\mathfrak{D}=q\frac{d}{dq}}\) on brackets vanishes on the MZV side, which gives rise to many nontrivial \({\mathbb{Q}}\)-linear relations.