Mathematische Annalen

, Volume 365, Issue 1–2, pp 345–362 | Cite as

Period polynomial relations between formal double zeta values of odd weight

Article

Abstract

For odd k, we give a formula for the relations between double zeta values \(\zeta (r,k-r)\) with r even. This formula provides a connection with the space of cusp forms on \(\mathrm {SL}_2(\mathbb Z)\). This is the odd weight analogue of a result in (Gangl et al., Double zeta values and modular forms, In: Bocherer, S., Ibukiyama, T., Kaneko, M., Sato, F. (eds.) Automorphic Forms and Zeta Functions, Proceedings of the Conference in Memory of Tsuneo Arakawa, pp 71–106, World Scientific, New Jersey, 2006) by Gangl, Kaneko and Zagier. We also provide an answer of a question asked by Zagier in (Ann. Math. 175:977–1000, 2006) about the left kernel of some matrix. Although the restricted sum statement in (Gangl et al., Double zeta values and modular forms, In: Bocherer, S., Ibukiyama, T., Kaneko, M., Sato, F. (eds.) Automorphic Forms and Zeta Functions, Proceedings of the Conference in Memory of Tsuneo Arakawa, pp 71–106, World Scientific, New Jersey, 2006) fails in the odd weight case, we provide an asymptotical statement that replaces it. Our statement works more generally for restricted sums with any congruence condition on the first entry of the double zeta value.

Mathematics Subject Classification

11F67 11M32 

Notes

Acknowledgments

This study was funded by NSF Grant No. DMS-1401122. The author would like to thank Romyar Sharifi for introducing him a project related to this area, David Broadhurst for his numerical data, Masanobu Kaneko and Herbert Gangl for very useful comments on the first draft.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA

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