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Zbl 1122.11057
Gangl, Herbert; Kaneko, Masanobu; Zagier, Don
Double zeta values and modular forms.
(English)
[A] Böcherer, Siegfried (ed.) et al., Automorphic forms and zeta functions. In memory of Tsuneo Arakawa. Proceedings of the conference, Rikkyo University, Tokyo, Japan, September 4--7, 2004. Hackensack, NJ: World Scientific. 71-106 (2006). ISBN 981-256-632-5/hbk

The double zeta values are defined for integers $r\geq 2,s\geq 1,$ by $\zeta(r,s)=\sum_{m>n>0}m^{-r}n^{-s}$. The present paper gives various interesting relations among double zeta values, e.g., $$28\zeta(9,3)+150\zeta(7,5)+168\zeta(5,7)={5197\over 691}\zeta(12).$$ It is shown that the structure of the $Q$-vector space of all relations among double zeta values of fixed weight $k=r+s$ is connected with the structure of the space of modular forms of weight $k$ on the full modular group (as indicated by the appearance of $691$ in the formula above). Moreover, the authors introduce both transcendental and combinatorial double Eisenstein series in order to study the relations between double zeta values and modular forms.
[Jörn Steuding (Würzburg)]
MSC 2000:
*11M41 Other Dirichlet series and zeta functions
11F11 Modular forms, one variable

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