Skip to main content
Log in

Uniform bound for HeckeL-functions

  • Published:
Acta Mathematica

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Bruggeman, R. W., Fourier coefficients of cusp forms.Invent. Math., 45 (1978), 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  2. Bruggeman, R. W. &Motohashi, Y., Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field.Funct. Approx. Comment. Math., 31 (2003), 23–92.

    MATH  MathSciNet  Google Scholar 

  3. — A new approach to the spectral theory of the fourth moment of the Riemann zeta-function.J. Reine Angew. Math., 579 (2005), 75–114.

    MATH  MathSciNet  Google Scholar 

  4. Good, A., The square mean of Dirichlet series associated with cusp forms.Mathematika, 29 (1982), 278–295.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ivić, A., On sums of Hecke series in short intervals.J. Théor. Nombres Bordeaux, 13 (2001), 453–468.

    MathSciNet  MATH  Google Scholar 

  6. Iwaniec, H., Fourier coefficients of cusp forms and the Riemann zeta-function, inSeminar on Number Theory, 1979–80, Exp. 18. Univ. Bordeaux I, Talence, 1980.

    Google Scholar 

  7. — Small eigenvalues of Laplacian for Γ0(N).Acta Arith., 56 (1990), 65–82.

    MATH  MathSciNet  Google Scholar 

  8. Jutila, M.,Lectures on a Method in the Theory of Exponential Sums. Tata Inst. Fund. Res. Lectures on Math. and Phys., 80. Springer, Berlin, 1987.

    MATH  Google Scholar 

  9. — The additive divisor problem and its analogs for Fourier coefficients of cusp forms, I.Math. Z., 223 (1996), 435–461;Ibid., The additive divisor problem and its analogs for Fourier coefficients of cusp forms, II., 225 (1997), 625–637.

    Article  MATH  MathSciNet  Google Scholar 

  10. — Mean values of Dirichlet series via Laplace transforms, inAnalytic Number Theory (Kyoto, 1996), pp. 169–207. Cambridge Univ. Press, Cambridge, 1997.

    Google Scholar 

  11. —, On spectral large sieve inequalities.Funct. Approx. Comment. Math., 28 (2000), 7–18.

    MATH  MathSciNet  Google Scholar 

  12. —, The fourth moment of central values of Hecke series, inNumber Theory (Turku, 1999), pp. 167–177. de Gruyter, Berlin, 2001.

    Google Scholar 

  13. —, The spectral mean square of HeckeL-functions on the critical line.Publ. Inst. Math. (Beograd) (N.S.), 76 (90) (2004), 41–55.

    MathSciNet  Google Scholar 

  14. Jutila, M. &Motohashi, Y., A note on the mean value of the zeta andL-functions, XI.Proc. Japan Acad. Ser. A Math. Sci., 78 (2002), 1–6.

    MATH  MathSciNet  Google Scholar 

  15. Katok, S. &Sarnak, P., Heegner points, cycles and Maass forms.Israel J. Math., 84 (1993), 193–227.

    MATH  MathSciNet  Google Scholar 

  16. Kuznetsov, N. V., The Petersson hypothesis for forms of weight zero and the Linnik hypothesis. Preprint, Khabarovsk Complex Res. Inst. Acad. Sci. USSR, 1977 (Russian).

  17. —, Convolution of Fourier coefficients of Eisenstein-Maass series.Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI), 129 (1983), 43–84 (Russian).

    MathSciNet  MATH  Google Scholar 

  18. Lebedev N. N.,Special Functions and Their Applications. Dover, New York, 1972.

    MATH  Google Scholar 

  19. Meurman, T., On the order of the MaassL-function on the critical line, inNumber Theory, Vol. I (Budapest, 1987), pp. 325–354. Colloq. Math. Soc. János Bolyai, 51. North-Holland, Amsterdam, 1990.

    Google Scholar 

  20. Motohashi, Y., An explicit formula for the fourth power mean of the Riemann zeta-function.Acta Math., 170 (1993), 181–220.

    Article  MATH  MathSciNet  Google Scholar 

  21. —, The binary additive divisor problem.Ann. Sci. École Norm. Sup., 27 (1994), 529–572.

    MATH  MathSciNet  Google Scholar 

  22. Motohashi, Y., The mean square of HeckeL-series attached to holomorphic cusp forms, inAnalytic Number Theory (Kyoto, 1993).Sûrikaisekikenkyûsho Kôkyûroku, 886 (1994), 214–227.

  23. —,Spectral Theory of the Riemann Zeta-Function, Cambridge Tracts in Math., 127. Cambridge Univ. Press, Cambridge, 1997.

    MATH  Google Scholar 

  24. —, A note on the mean value of the zeta andL-functions, XIV.Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), 28–33.

    Article  MATH  MathSciNet  Google Scholar 

  25. Sarnak, P., Estimation of Rankin-SelbergL-functions and quantum unique ergodicity.J. Funct. Anal., 184 (2001), 419–453.

    Article  MATH  MathSciNet  Google Scholar 

  26. Titchmarsh, E. C.,The Theory of the Riemann Zeta-Function. Oxford Univ. Press, Oxford, 1951.

    MATH  Google Scholar 

  27. Watson, G. N.,A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge, 1995.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The first author was supported by the grant 8205966 from the Academy of Finland, and the second author by KAKENHI 15540047 and Nihon University research grant (2004).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jutila, M., Motohashi, Y. Uniform bound for HeckeL-functions. Acta Math. 195, 61–115 (2005). https://doi.org/10.1007/BF02588051

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02588051

Keywords

Navigation