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Mixed Tate motives and multiple zeta values. (English) Zbl 1042.11043
The author establishes a striking result on Zagier’s conjecture on the mulitple zeta values. Let $$L_n$$ be the subspace of $$\mathbb{R}$$ generated by the multiple zeta values of weight $$n$$ and let $$d_n$$ be the positive integer defined by $$\sum_{i=1}^\infty d_it^i= (1-t^2-t^3)^{-1}$$. Zagier’s conjecture then asserts the equality $$\dim(L_n)= d_n$$. The main result of this paper is to show the inequality $$\dim (L_n)\leq d_n$$. The outline of the proof is as follows: Firstly, the multiple zeta values of weight $$n$$, which are given by the iterated integrals, can be interpreted as the periods of the relative cohomology $$H^n$$ of certain periods of algebraic varieties. Next, regard $$H^n$$ as an object in the mixed Tate motives in which the dimensions of the extension groups can be estimated by the known results on algebraic $$K$$-groups. This yields the desired inequality. The same result was independently obtained by A. B. Goncharov in a different way [European Congress of Mathematics, Vol. I, 361–392 (2001; Zbl 1042.11042), see the preceding review].

MSC:
 11G55 Polylogarithms and relations with $$K$$-theory 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 11M41 Other Dirichlet series and zeta functions
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