Double zeta values and modular forms. (English) Zbl 1122.11057

Gangl, Herbert; Kaneko, Masanobu; Zagier, Don

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 \(r\geq 2,s\geq 1,\) 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.
For the entire collection see [Zbl 1091.11003].

Reviewer: Jörn Steuding (Würzburg)

MSC:

11M32	Multiple Dirichlet series and zeta functions and multizeta values
11F11	Holomorphic modular forms of integral weight