Skip to main content
SpringerLink
Log in

Analogues of the Hurwitz Formulas for Level 2 Eisenstein Series

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider certain double series of Eisenstein type involving hyperbolic functions, which can be regarded as analogues of the level 2 Eisenstein series. We prove some evaluation formulas for these series at positive integers which are analogues of both the Hurwitz formulas for the level 2 Eisenstein series and the classical results given by Cauchy, Lerch, Mellin and Ramanujan.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ayoub R.: Euler and the zeta function. Am. Math. Mon. 81, 1067–1086 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berndt B.C.: Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7, 147–189 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berndt B.C.: Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew Math. 303/304, 332–365 (1978)

    MathSciNet  Google Scholar 

  4. Berndt B.C.: Ramanujan’s Notebooks, Part II. Springer-Verlag, New York (1989)

    MATH  Google Scholar 

  5. Berndt B.C.: Ramanujan’s Notebooks, Part V. Springer-Verlag, New York (1998)

    MATH  Google Scholar 

  6. Cauchy, A.: Oeuvres Completes D’Augustin Cauchy, Série II, t. VII. Gauthier-Villars, Paris (1889)

  7. Hurwitz, A.: Mathematische Werke. Bd. I: Funktionentheorie. (German) Herausgegeben von der Abteilung für Mathematik und Physik der Eidgenössischen Technischen Hochschule in Zürich. Birkhäuser Verlag, Basel-Stuttgart (1962)

  8. Katayama K.: On the values of Eisenstein series. Tokyo J. Math. 1, 157–188 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  9. Koblitz, N.: Introduction to elliptic curves and modular forms, 2nd edn. In: Graduate Texts in Mathematics, No. 97. Springer-Verlag, New York (1993)

  10. Komori, Y., Matsumoto, K., Tsumura, H.: Hyperbolic-sine analogues of Eisenstein series, generalized Hurwitz numbers, and q-zeta functions. Preprint, arXiv:math1006.3339

  11. Lemmermeyer F.: Reciprocity Laws: From Euler to Eisenstein. Springer, Berlin (2000)

    MATH  Google Scholar 

  12. Lerch M.: Sur la fonction ζ (s) pour valeurs impaires de l’argument. J. Sci. Math. Astron. pub. pelo Dr. F. Gomes Teixeira, Coimbra 14, 65–69 (1901)

    Google Scholar 

  13. Mellin H.J.: Eine Formel für den Logarithmus transcendenter Funktionen von endlichem Geschlecht. Acta Soc. Sci. Fennicae 29(4), 49 (1902)

    MathSciNet  Google Scholar 

  14. Mellin H.J.: Eine Formel für den Logarithmus transcendenter Funktionen von endlichem Geschlecht. Acta. Math. 25(1), 165–183 (1902)

    Article  MathSciNet  Google Scholar 

  15. Malurkar S.L.: On the application of Herr Mellin’s integrals to some series. J. Indian Math. Soc. 16, 130–138 (1925/26)

    Google Scholar 

  16. Nanjundiah T.S.: Certain summations due to Ramanujan, and their generalizations. Proc. Indian Acad. Sci. Sec. A 34, 215–228 (1951)

    MATH  MathSciNet  Google Scholar 

  17. Phillips E.G.: Note on summation of series. J. Lond. Math. Soc. 4, 114–116 (1929)

    Article  Google Scholar 

  18. Prasolov, V., Solovyev, Y.: Elliptic functions and elliptic integrals (translated from the Russian manuscript by D. Leites). Translations of Mathematical Monographs, 170. American Mathematical Society, Providence (1997)

  19. Rieger G.J.: Eine Bemerkung über die Hurwitzschen Zahlen. J. Reine Angew. Math. 296, 212–216 (1977)

    MATH  MathSciNet  Google Scholar 

  20. Serre, J.-P.: A course in arithmetic. In: Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York (1973)

  21. Tsumura H.: On some combinatorial relations for Tornheim’s double series. Acta Arith. 105, 239–252 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Tsumura H.: On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function. Math. Proc. Camb. Philos. Soc. 142, 395–405 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tsumura H.: On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type. Bull. Lond. Math. Soc. 40, 85–93 (2008)

    Article  MathSciNet  Google Scholar 

  24. Tsumura H.: Evaluation of certain classes of Eisenstein type series. Bull. Austral. Math. Soc. 79, 239–247 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Watson G.N.: Theorems stated by Ramanujan II. J. Lond. Math. Soc. 3, 216–225 (1929)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Hirofumi Tsumura

Authors
  1. Hirofumi Tsumura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hirofumi Tsumura.

Additional information

This research was partially supported by Grand-in-Aid for Science Research (No. 20540020), Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tsumura, H. Analogues of the Hurwitz Formulas for Level 2 Eisenstein Series. Results. Math. 58, 365–378 (2010). https://doi.org/10.1007/s00025-010-0058-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-010-0058-9

Mathematics Subject Classification (2000)

Keywords

Search

Navigation