Skip to main content
SpringerLink
Log in

Ramanujan identities of higher degree

  • Research
  • Published:
Research in Number Theory Aims and scope Submit manuscript

Abstract

We use techniques regarding generalized Dirichlet series developed in Franke (Ramanujan J 46(1):91–102, 2018) to obtain formulas for a wide class of L-functions at rational arguments. It is shown that these values are related to special functions on the upper half plane which possess similar properties as modular forms. Several formulas of Ramanujan involving values of L-functions at integer arguments turn out to be special cases of the main theorem.

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. Bateman, H.: Tables of Integral Transforms, vol. 1. McGraw-Hill Book Company, New York (1954)

    Google Scholar 

  2. Berndt, B.: Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7(1), 147–189 (1977)

    Article  MathSciNet  Google Scholar 

  3. Berndt, B., Straub, A.: On a secant Dirichlet series and Eichler integrals of Eisenstein series. Math. Z. 284(3), 827–852 (2016)

    Article  MathSciNet  Google Scholar 

  4. Bochner, S.: Some properties of modular relations. Ann. Math. 53, 332–363 (1951)

    Article  MathSciNet  Google Scholar 

  5. Bruggeman, R., Choie, Y.J., Diamantis, N.: Holomorphic automorphic forms and cohomology. arXiv: 1404.6718 (2017)

  6. Franke, J.: Infinite series representations for Dirichlet \(L\)-functions at rational arguments. Ramanujan J. 46(1), 91–102 (2018)

    Article  MathSciNet  Google Scholar 

  7. Gun, S., Murty, M.R., Rath, P.: Transcendental values of certain Eichler integrals. Bull. Lond. Math. Soc. 43(5), 939–952 (2011)

    Article  MathSciNet  Google Scholar 

  8. Hecke, E.: Über die Zetafunktion beliebiger algebraischer Zahlkörper, Mathematische Werke Nr. 7, 159–171. Vandenhoeck & Ruprecht, Göttingen (1970)

  9. Jin, S., Ma, W., Ono, K., Soundararajan, K.: The Riemann hypothesis for period polynomials of modular forms. Proc. Natl. Acad. Sci. USA 113(10), 2603–2608 (2016)

    Article  MathSciNet  Google Scholar 

  10. Kohnen, W., Raji, W.: Special values of Hecke \(L\)-functions of modular forms of half-integral weight and cohomology. Res. Math. Sci. 5, 22 (2018)

    Article  MathSciNet  Google Scholar 

  11. Lewis, J., Zagier, D.: Period functions for Maass wave forms. Ann. Math. Second Ser. 153(1), 191–258 (2001)

    Article  MathSciNet  Google Scholar 

  12. Miyake, T.: Modular Forms, Springer Monographs in Mathematics. Springer, New York (1989)

    Google Scholar 

Download references

Author's contributions

The author is very grateful to Winfried Kohnen for many helpful comments as well as the referees for their careful reading and their useful suggestions which improved the paper.

Author information

Authors and Affiliations

  1. Mathematisches Institut Universität Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    J. Franke

Authors
  1. J. Franke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Franke.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Franke, J. Ramanujan identities of higher degree. Res. number theory 4, 42 (2018). https://doi.org/10.1007/s40993-018-0135-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40993-018-0135-9

Keywords

Mathematics Subject Classification

Search

Navigation