Article

Derivation and double shuffle relations for multiple zeta values

Collège de France, Lutetia Parisorum, Île-de-France, France
Compositio Mathematica (Impact Factor: 0.99). 03/2006; 142(02). DOI: 10.1112/S0010437X0500182X
Source: OAI

ABSTRACT

Derivation and extended double shuffle (EDS) relations for multiple zeta values (MZVs) are proved. Related algebraic structures of MZVs, as well as a version of EDS relations are also studied.

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    • "We note that ∂ l (1) = 0. In [3], K. Ihara, M. Kaneko and D. Zagier proved the derivation relation for MZVs. "
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    ABSTRACT: The derivation relation for multiple zeta values is proved by Ihara, Kaneko and Zagier. We prove its counterpart for finite multiple zeta values.
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    • "We will focus on quasi-shuffle relations, directly derived from the nested sum representation (2), for example, ζ(a)ζ(b) = ζ(a, b) + ζ(b, a) + ζ(a + b). The two representations combined yield intricate relations, which are commonly refereed to as double shuffle structures underlying MZVs [10]. "
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    ABSTRACT: We describe in this work all solutions to the problem of renormalizing multiple zeta values at arguments of any sign in a quasi-shuffle compatible way. As a corollary we clarify the relation between different renormalizations at non-positive values appearing in the recent literature.
    Full-text · Article · Oct 2015
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    • "Several of the papers mentioned above also contain formulas of a similar flavor for the values of more general " multiple zeta values " defined in terms of several integer-valued parameters; see also [14], [20], [29], [45] for related results. The results described above may not make evident why it is natural to consider ω(s) as a true Dirichlet series (that is, as a function of a complex variable s), but plenty of precedents from the history of analytic number theory suggest that this is worth doing. "
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    ABSTRACT: We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number of n-dimensional representations of SU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.
    Preview · Article · Mar 2015
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