Differential algebraic Birkhoff decomposition and the renormalization of multiple zeta values. (English) Zbl 1186.11052
This paper, together with the companion paper [J. Algebra 319, No. 9, 3770–3809 (2008; Zbl 1165.11071)], gives a new approach to the problem of defining multiple zeta values in such a way that they would be defined at nonpositive integers, and at the same time the definition would agree with the analytic continuation approach. The authors propose an adaptation of the rigorous renormalization methods from quantum field theory developed by A. Connes and D. Kreimer [Commun. Math. Phys. 210, No. 1, 249–273 (2000; Zbl 1032.81026); Commun. Math. Phys. 216, No. 1, 215–241 (2001; Zbl 1042.81059)]. A key element is a reformulation of the above method in terms of differential algebra (differential Hopf algebras and differential Rota-Baxter algebras). For an extension to the case of \(q\)-zeta function see [J. Zhao, Acta Math. Sin., Engl. Ser. 24, No. 10, 1593–1616 (2008; Zbl 1186.11054)].
Reviewer: Anatoly N. Kochubei (Kyïv)
MSC:
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
| 12H05 | Differential algebra |
| 16S99 | Associative rings and algebras arising under various constructions |
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