Skip to main content
SpringerLink
Log in

Finite trigonometric sums and class numbers

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

Explicit evaluations of finite trigonometric sums arose in proving certain theta function identities of Ramanujan. In this paper, without any appeal to theta functions, several classes of finite trigonometric sums, including the aforementioned sums, are evaluated in closed form in terms of class numbers of imaginary quadratic fields.

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. Apostol, T.M.: Euler’s φ-function and separable Gauss sums. Proc. Am. Math. Soc. 24, 482–485 (1970)

    MATH  Google Scholar 

  2. Ayoub, R.: An Introduction to the Analytic Theory of Numbers. American Mathematical Society, Providence, RI, 1963

  3. Berndt, B.C.: Classical theorems on quadratic residues. Enseign. Math. 22, 261–304 (1976)

    MATH  Google Scholar 

  4. Berndt, B.C.: Ramanujan’s Notebooks, Part IV. Springer-Verlag, New York, 1994

  5. Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Wiley, New York, 1998

  6. Berndt, B.C., Zhang, L.-C.: Ramanujan’s identities for eta-functions. Math. Ann. 292, 561–573 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Borevich, Z.I., Shafarevich, I.R.: Number Theory. Academic Press, New York, 1966

  8. Chowla, S.: The evaluation of a trigonometric sum. J. Indian Math. Soc. 18, 147–148 (1929)

    MATH  Google Scholar 

  9. Chowla, S.: The Collected Papers of Sarvadaman Chowla. Vol. I, Les Publications CRM, Montreal, 1999

  10. Davenport, H.: Multiplicative Number Theory. 3rd ed., Springer-Verlag, New York, 2000

  11. Dickson, L.E.: History of the Theory of Numbers. Vol. 3, Chelsea, New York, 1966

  12. Dirichlet, G.L.: Recherches sur diverses applications de l’analyse infinitésimale à le théorie des nombres, seconde partie. J. Reine Angew. Math. 21, 134–155 (1840)

    Google Scholar 

  13. Lebesgue, V.A.: Suite du Memoire sur les applications du symbole J. de Math. 15, 215–237 (1850)

    Google Scholar 

  14. Lerch, M.: Essais sur le calcul du nombre des classes de formes quadratiques binaires aux coefficients entiers. Acta Math. 30, 203–293 (1906)

    MATH  Google Scholar 

  15. Liu, Z.-G.: Some Eisenstein series identities related to modular equations of the seventh order. Pacific J. Math. 209, 103–130 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Ramanujan, S.: Notebooks. (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957

  17. Schemmel, V.: De Multitudine formarum secundi gradus disquisitiones. Dissertation, Breslau, 1863

  18. Stoneham, R.G.: Letter to R. J. Evans. September 3, 1982

  19. Whiteman, A.L.: Theorems on quadratic residues. Math. Magazine 23, 71–74 (1949)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801, USA

    Bruce C. Berndt & Alexandru Zaharescu

Authors
  1. Bruce C. Berndt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandru Zaharescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruce C. Berndt.

Additional information

Mathematics Subject Classification (2000): Primary, 11L03; Secondary, 11R29, 11L10

Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.

Revised version: 19 April 2004

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berndt, B., Zaharescu, A. Finite trigonometric sums and class numbers. Math. Ann. 330, 551–575 (2004). https://doi.org/10.1007/s00208-004-0559-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-004-0559-5

Keywords

Search

Navigation