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Double zeta values and modular forms. (English) Zbl 1122.11057
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 (ISBN 981-256-632-5/hbk). 71-106 (2006).

The double zeta values are defined for integers r2,s1, by ζ(r,s)= m>n>0 m -r n -s . The present paper gives various interesting relations among double zeta values, e.g.,

28ζ(9,3)+150ζ(7,5)+168ζ(5,7)=5197 691ζ(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.

11M41Other Dirichlet series and zeta functions
11F11Holomorphic modular forms of integral weight