Proceedings of the Japan Academy, Series A, Mathematical Sciences

Some restricted sum formulas for double zeta values

Tomoya Machide

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We give some restricted sum formulas for double zeta values whose arguments satisfy certain congruence conditions modulo 2 or 6, and also give an application to identities showed by Ramanujan for sums of products of Bernoulli numbers with a gap of 6.

Article information

Proc. Japan Acad. Ser. A Math. Sci. Volume 89, Number 3 (2013), 51-54.

First available in Project Euclid: 1 March 2013

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Zentralblatt MATH identifier

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11B68: Bernoulli and Euler numbers and polynomials

Multiple zeta value double zeta value sum formula Bernoulli number Ramanujan’s identity


Machide, Tomoya. Some restricted sum formulas for double zeta values. Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 3, 51--54. doi:10.3792/pjaa.89.51.

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  • B. C. Berndt, Ramanujan's notebooks. Part I, Springer, New York, 1985.
  • L. Euler, Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol. 20 (1775), 140–186, reprinted in Opera omnia, ser. I, vol. 15, Teubner, Berlin, 1927, pp. 217–267.
  • H. Gangl, M. Kaneko and D. Zagier, Double zeta values and modular forms, in Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, pp. 71–106.
  • T. Nakamura, Restricted and weighted sum formulas for double zeta values of even weight, Šiauliai Math. Semin. 4(12) (2009), 151–155.
  • Y. Ohno and W. Zudilin, Zeta stars, Commun. Number Theory Phys. 2 (2008), no. 2, 325–347.
  • S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc. 3 (1911), 219–234, reprinted in Collected papers of Srinivasa Ramanujan, Cambridge Univ. Press, Cambridge, 1927, pp. 1–14.
  • S. S. Wagstaff, Jr., Ramanujan's paper on Bernoulli numbers, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1–4, 49–65.