Renormalization of multiple zeta values

Abstract Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics and physics with broad connections and applications. In contrast, very little is known about the special values of multiple zeta functions at non-positive integers since the values are usually undefined. We define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory, and following the Hopf algebra approach of Connes and Kreimer. This definition of renormalized MZVs agrees with the convergent MZVs and extends the work of Ihara–Kaneko–Zagier on renormalization of MZVs with positive arguments. We further show that the important quasi-shuffle (stuffle) relation for usual MZVs remains true for the renormalized MZVs.

[1]  Equivariant Todd Classes for Toric Varieties , 2003, math/0311318.

[2]  D. Bradley Multiple $q$-Zeta Values , 2004, math/0402093.

[3]  Jianqiang Zhao,et al.  ANALYTIC CONTINUATION OF MULTIPLE ZETA FUNCTIONS , 1999 .

[4]  Michael E. Hoffman,et al.  Multiple harmonic series. , 1992 .

[5]  R. Ehrenborg On Posets and Hopf Algebras , 1996 .

[6]  Le Bois-Marie,et al.  Integrable Renormalization II : the general case , 2004 .

[7]  Li Guo,et al.  Differential Algebraic Birkhoff Decomposition and the renormalization of multiple zeta values , 2008 .

[8]  Masanobu Kaneko,et al.  Multiple zeta values, poly-Bernoulli numbers, and related zeta functions , 1999, Nagoya Mathematical Journal.

[9]  D. Manchon Hopf algebras, from basics to applications to renormalization , 2004, math/0408405.

[10]  Li Guo,et al.  Baxter Algebras and Hopf Algebras , 2003 .

[11]  Li Guo,et al.  Multiple zeta values and Rota--Baxter algebras , 2006 .

[12]  Takashi Aoki,et al.  Zeta Functions, Topology and Quantum Physics , 2005 .

[13]  井原 健太郎,et al.  Derivation and double shuffle relations for multiple zeta values: joint work with M.Kaneko, D.Zagier (多重ゼータ値の研究--短期共同研究報告集) , 2007 .

[14]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem , 2000 .

[15]  Michael E. Hoffman,et al.  The Algebra of Multiple Harmonic Series , 1997 .

[16]  Sylvie Paycha,et al.  Shuffle Relations for Regularised Integrals of Symbols , 2005 .

[17]  Jonathan M. Borwein,et al.  Special values of multiple polylogarithms , 1999, math/9910045.

[18]  Li Guo,et al.  Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion , 2006, math-ph/0602004.

[19]  Samuel K. Hsiao,et al.  Canonical Characters on Quasi-Symmetric Functions and Bivariate Catalan Numbers , 2005, Electron. J. Comb..

[20]  D. Kreimer Knots and Feynman Diagrams , 2000 .

[21]  Li Guo,et al.  Mixable Shuffles, Quasi-shuffles and Hopf Algebras , 2008 .

[22]  Li Guo Baxter Algebras, Stirling Numbers and Partitions , 2004 .

[23]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[24]  Michiel Hazewinkel,et al.  Generalized Overlapping Shuffle Algebras , 2000 .

[25]  Kohji Matsumoto The Analytic Continuation and the Asymptotic Behaviour of Certain Multiple Zeta-Functions(III) , 2003 .

[26]  Li Guo,et al.  On Free Baxter Algebras: Completions and the Internal Construction , 2000 .

[27]  Combinatorics of renormalization as matrix calculus , 2005, hep-th/0508154.

[28]  A. Goncharov Periods and mixed motives , 2002, math/0202154.

[29]  Jianqiang Zhao Renormalization of multiple q-zeta values , 2008 .

[30]  Alain Connes,et al.  Renormalization in quantum field theory and the Riemann-Hilbert problem , 1999 .

[31]  Kohji Matsumoto Functional equations for double zeta-functions , 2004, Mathematical Proceedings of the Cambridge Philosophical Society.

[32]  Michael E. Hoffman,et al.  Quasi-Shuffle Products , 1999 .

[33]  Hector Figueroa,et al.  Combinatorial Hopf algebras in quantum field theory. I , 2005 .

[34]  Li Guo,et al.  Spitzer's identity and the algebraic Birkhoff decomposition in pQFT , 2004, hep-th/0407082.

[35]  D. Zagier Values of Zeta Functions and Their Applications , 1994 .

[36]  Pierre Cartier,et al.  Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents , 2001 .

[37]  Jean-Louis Loday On the algebra of quasi-shuffles , 2005 .

[38]  S. Akiyama,et al.  Multiple Zeta Values at Non-Positive Integers , 2001 .

[39]  Pierre Cartier,et al.  On the structure of free baxter algebras , 1972 .

[40]  U MichaelE.Hoffman The Algebra of Multiple Harmonic Series , 1997 .

[41]  Li Guo,et al.  Integrable renormalization I: The ladder case , 2004, hep-th/0402095.

[42]  F. Spitzer A Combinatorial Lemma and its Application to Probability Theory , 1956 .

[43]  Gian-Carlo Rota,et al.  Baxter algebras and combinatorial identities. II , 1969 .

[44]  Li Guo,et al.  Baxter Algebras and Shuffle Products , 2000 .

[45]  Kohji Matsumoto Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series , 2003, Nagoya Mathematical Journal.

[46]  Shigeki Akiyama,et al.  Analytic continuation of multiple zeta-functions and their values at non-positive integers , 2001 .

[47]  Tomohide Terasoma,et al.  Mixed Tate motives and multiple zeta values , 2002 .

[48]  Li Guo Baxter Algebras and the Umbral Calculus , 2001, Adv. Appl. Math..

[49]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group , 2001 .

[50]  Doubles mélanges des polylogarithmes multiples aux racines de l’unité , 2002, math/0202142.

[51]  Michael E. Hoffman Algebraic Aspects of Multiple Zeta Values , 2003, math/0309425.

[52]  Partition Identities for the Multiple Zeta Function , 2004, math/0402091.

[53]  D. J. Broadhurst,et al.  Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops , 1996, hep-th/9609128.

[54]  Li Guo,et al.  Operated semigroups, Motzkin paths and rooted trees , 2007, 0710.0429.

[55]  Matilde Marcolli,et al.  Renormalization, the Riemann–Hilbert Correspondence, and Motivic Galois Theory , 2004, hep-th/0411114.

[56]  A. Goncharov,et al.  Multiple $\zeta$-motives and moduli spaces $\overline{\mathcal{M}}_{0,n}$ , 2004, Compositio Mathematica.

[57]  David M. Bradley,et al.  Multiple Zeta Values , 2005 .