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Transformation formula for exponential sums involving fourier coefficients of modular forms

  • C. S. Yogananda
Article

Abstract

In 1984 Jutila [5] obtained a transformation formula for certain exponential sums involving the Fourier coefficients of a holomorphic cusp form for the full modular groupSL(2, ℤ). With the help of the transformation formula he obtained good estimates for the distance between consecutive zeros on the critical line of the Dirichlet series associated with the cusp form and for the order of the Dirichlet series on the critical line, [7]. In this paper we follow Jutila to obtain a transformation formula for exponential sums involving the Fourier coefficients of either holomorphic cusp forms or certain Maass forms for congruence subgroups ofSL(2, ℤ) and prove similar estimates for the corresponding Dirichlet series.

Keywords

Exponential sums summation formula cusp forms and Maass forms transformation formula 

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References

  1. [1]
    Balasubramanian R, An improvement of a theorem of Titchmarsh on the mean square of ¦ζ(l/2 +it)¦,Proc. London Math. Soc. 36 (1978) 540–576zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Berndt B C, The Voronoi summation formula inThe theory of arithmetic functions 21–36. Lecture Notes in Mathematics 251, (Springer) (1972)Google Scholar
  3. [3]
    Berndt B C, Identities involving the coefficients of a class of Dirichlet series, V,Trans. Am. Math. Soc. 160 (1971) 139–156zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Good A, The square mean of Dirichlet series associated with cusp forms,Mathematika 29 (1982) 278–295zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Jutila M, Transformation formulae for Dirichlet polynomials,J. Number Theory 18 (1984) 135–156zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Jutila M, Lectures on a Method in the Theory of Exponential Sums. Tata Institute of Fundamental Research, Bombay 1987.zbMATHGoogle Scholar
  7. [7]
    Jutila M, The fourth power moment of the Riemann Zeta function over a short interval.Proc. Coll. Janos Bolyai, Coll. Number Theory, Budapest (1987).Google Scholar
  8. [8]
    Meurman T, On exponential sums involving the Fourier coefficients of Maass wave forms,J. Reine Angew. Math. 384 (1988) 192–207zbMATHMathSciNetGoogle Scholar
  9. [9]
    Meurman T, On the order of the MaassL-functions on the critical line,Proc. Coll. Janos Bolyai, Coll. Number Theory, Budapest (1987).Google Scholar
  10. [10]
    Ogg A,Modular forms and Dirichlet series (New York: Benjamin) (1969)zbMATHGoogle Scholar
  11. [11]
    Terras A,Harmonic analysis on symmetric spaces and applications (Springer-Verlag) (1985)Google Scholar
  12. [12]
    Razar M J, Modular forms for Гo(N) and Dirichlet series,Trans. Am. Math. Soc. 231 (1977) 489–495zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Weil A, Über die Bestimmung Dirichletcher Reihen durch Funktionalgleichungen,Math. Ann. 168 (1967) 149–156zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 1993

Authors and Affiliations

  • C. S. Yogananda
    • 1
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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