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 singular. We define and study multiple zeta functions at any 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.

36 pages, revised the layout and corrected typos

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Cited by in corpus (10)

• Rota-Baxter Algebras and Dendriform Algebras
• Free Rota-Baxter algebras and rooted trees
• H = x p with interaction and the Riemann zeros
• Renormalization of Multiple $q$-Zeta Values
• Differential Birkhoff decomposition and the renormalization of multiple zeta values
• Representations of Rota-Baxter algebras and regular singular decompositions
• On the Quantum Reconstruction of the Riemann zeros
• Nested sums of symbols and renormalised multiple zeta functions
• Operated semigroups, Motzkin paths and rooted trees
• On Differential Rota-Baxter Algebras