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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 √Čtale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11M41 Other Dirichlet series and zeta functions
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