Advertisement

Integral Analogues of Theta Functions and Gauss Sums

  • George E. Andrews
  • Bruce C. Berndt
Chapter

Abstract

This chapter is also related to two early papers on integrals, but all of the results are new. The integral examined by Ramanujan in this chapter satisfies a transformation formula similar to that satisfied by the classical theta functions. The integral can also be thought of as an analogue of Gauss sums or as an analogue of the classical Weierstrass σ-function.

Keywords

Theta Function Transformation Formula Reciprocity Theorem Integral Analogue Foregoing Equality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 54.
    B.C. Berndt, R.J. Evans, and K.S. Williams, Gauss and Jacobi Sums, John Wiley, New York, 1998.MATHGoogle Scholar
  2. 69.
    B.C. Berndt and P. Xu, An integral analogue of theta functions and Gauss sums in Ramanujan’s lost notebook, Math. Proc. Cambridge Philos. Soc. 147 (2009), 257–265.CrossRefMathSciNetMATHGoogle Scholar
  3. 88.
    K. Chandrasekharan, Elliptic Functions, Springer-Verlag, Berlin, 1985.CrossRefMATHGoogle Scholar
  4. 126.
    I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, San Diego, 1994.Google Scholar
  5. 192.
    N.S. Koshliakov, On an extension of some formulae of Ramanujan, Proc. London Math. Soc. 12 (1936), 26–32.CrossRefMathSciNetGoogle Scholar
  6. 194.
    N.S. Koshliakov (under the name N.S. Sergeev), A study of a class of transcendental functions defined by the generalized Riemann equation (in Russian), Trudy Mat. Inst. Steklov, Moscow, 1949.Google Scholar
  7. 199.
    E. Krätzel, Bemerkungen zu einem Gitterpunktsproblem, Math. Ann. 179 (1969), 90–96.CrossRefMathSciNetMATHGoogle Scholar
  8. 210.
    Y. Lee, Email to B.C. Berndt, August 1, 2008.Google Scholar
  9. 247.
    S. Ramanujan, Question 463, J. Indian Math. Soc. 5 (1913), 120.Google Scholar
  10. 256.
    S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 (1915), 75–85.Google Scholar
  11. 258.
    S. Ramanujan, Some definite integrals, J. Indian Math. Soc. 11 (1915), 81–87.Google Scholar
  12. 267.
    S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence, RI, 2000.Google Scholar
  13. 269.
    S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.MATHGoogle Scholar
  14. 306.
    E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1951.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • George E. Andrews
    • 1
  • Bruce C. Berndt
    • 2
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations