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Monatshefte für Mathematik

, Volume 164, Issue 2, pp 133–156 | Cite as

Transformation formulas associated with integrals involving the Riemann Ξ-function

  • Atul DixitEmail author
Article

Abstract

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β), where αβ = 1. This gives a unified approach for generating certain modular transformation formulas, including a famous formula of Ramanujan and Guinand.

Keywords

Riemann Ξ-function Hurwitz zeta function Modified Bessel function Residue theorem Mellin transform Ramanujan 

Mathematics Subject Classification (2000)

Primary 11M06 Secondary 11M35 

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References

  1. 1.
    Berndt, B.C.,Dixit A.: A transformation formula involving the Gamma and Riemann zeta functions in Ramanujan’s Lost Notebook. In: Alladi, K., Klauder, J., Rao, C.R. (eds.) The Legacy of Alladi Ramakrishnan in the Mathematical Sciences. Springer, New York (2010, to appear)Google Scholar
  2. 2.
    Berndt, B.C., Lee, Y., Sohn, J.: In: Alladi, K. (ed.) The formulas of Koshliakov and Guinand in Ramanujan’s lost notebook, Surveys in Number Theory, Series: Developments in Mathematics, vol. 17, pp. 21–42. Springer, New York (2008)Google Scholar
  3. 3.
    Bump D.: Autmorphic forms and representations. Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1997)Google Scholar
  4. 4.
    Dixit, A.: Analogues of a transformation formula of Ramanujan (submitted)Google Scholar
  5. 5.
    Dixit, A.: Transformation formulas associated with integrals involving the Riemann Ξ-function and Ramanujan’s cubic theory of elliptic functions, Ph.D. thesis, University of Illinois at Urbana-Champaign (2011)Google Scholar
  6. 6.
    Dixit, A.: Series transformations and integrals involving the Riemann Ξ-function. J. Math. Anal. Appl. (to appear)Google Scholar
  7. 7.
    Goldfeld, D.: Automorphic forms and L-functions for the group \({{GL}(n,\mathbb R)}\). Cambridge Studies in Advanced Mathematics, vol. 99. Cambridge University Press, Cambridge (2006)Google Scholar
  8. 8.
    Oberhettinger F.: Tables of Mellin Transforms. Springer, New York (1974)zbMATHGoogle Scholar
  9. 9.
    Gradshteyn, I.S., Ryzhik, I.M. (eds): Table of Integrals, Series, and Products, 5th edn. Academic Press, San Diego (1994)zbMATHGoogle Scholar
  10. 10.
    Guinand A.P.: Some rapidly convergent series for the Riemann ξ-function. Q. J. Math. (Oxford) 6, 156–160 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Magnus W., Oberhettinger F., Soni R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin (1966)zbMATHGoogle Scholar
  12. 12.
    Ramanujan S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)zbMATHGoogle Scholar
  13. 13.
    Ramanujan S.: New expressions for Riemann’s functions ξ(s) and Ξ(t). Q. J. Math. 46, 253–260 (1915)zbMATHGoogle Scholar
  14. 14.
    Titchmarsh E.C.: The Theory of the Riemann Zeta Function. Clarendon Press, Oxford (1986)zbMATHGoogle Scholar
  15. 15.
    Watson G.N.: Some self-reciprocal functions. Q. J. Math. (Oxford) 2, 298–309 (1931)CrossRefGoogle Scholar
  16. 16.
    Watson G.N.: Theory of Bessel Functions, 2nd edn. University Press, Cambridge (1966)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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