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Estimates for coefficients of L-functions. III

  • W. Duke
  • H. Iwaniec
Part of the Progress in Mathematics book series (PM, volume 102)

Abstract

In this sequence of papers we investigate Dirichlet series
$${\rm A}(s,X) = \sum\limits_1^\infty {{a_n}} X(n){n^{ - 3}}$$
(1)
having Euler products and compatible functional equations with the aim of estimating the coefficients a n . It was shown in [2], [3] by different techniques that if the analytic continuation and the functional equations hold for sufficiently many characters then a suitable upper bound for a n is true which is considerably better than that resulting from the absolute convergence of the series. In this installment we combine both techniques to give new results and improve on those of [3] when A(s, X) has an Euler product of degree 3.

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References

  1. [1]
    P. Deligne. — La conjecture de Weil I, Publ. Math. LH.E.S. 43, (1974), 273–307.MathSciNetCrossRefGoogle Scholar
  2. [2]
    W. Duke and H. Iwaniec. — Estimates for coefficients of L-functions. I in Automorphic Forms and Analytic Number Theory, CRM Publications, Montreal 1990, 43–47.Google Scholar
  3. [3]
    W. Duke and H. Iwaniec. — Estimates for coefficients of L-functions. II (to appear in the Proceedings of Amalfi Conference, 1989 ).Google Scholar
  4. [4]
    A. Selberg. — On the estimation of Fourier coefficients of modular forms, A.M.S. Proc. Symp. Pure Math. Vol. VIII, (1965), 1–15.Google Scholar
  5. [5]
    E.C. Titchmarsh. — The theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1951.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • W. Duke
    • 1
  • H. Iwaniec
    • 1
  1. 1.Department of Mathematics Hill Center for the Mathematical SciencesRutgers UniversityNew BrunswickUSA

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