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References

  1. Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien,Com. Math. Phys.,102 (1985), 497–502.

    Article  MATH  Google Scholar 

  2. J.-M. Deshouillers, H. Iwaniec, Kloosterman sum and Fourier coefficients of cusp forms,Invent. Math.,70 (1982), 219–288.

    Article  MATH  Google Scholar 

  3. J.-M. Deshouillers, H. Iwaniec, The non-vanishing of Rankin-Selberg zeta-functions at special points, Selberg trace formula and related topics,Contemp. Math.,53, Amer. Math. Soc., Providence, RI, 1986, 59–95.

    Google Scholar 

  4. A. Erdélyi et al., Higher Transcendental Functions, vol. 2, McGraw-Hill, 1953.

  5. I. S. Gradshtein, I. M. Ryzhik,Tables of Integrals, Series and Products, Academic Press, New York and London, 1965.

    Google Scholar 

  6. D. A. Hejhal, The Selberg Trace Formula for PSL(2,R), Vol. 1,Springer Lecture Notes,548, Springer-Verlag, 1976.

  7. D. A. Hejhal, The Selberg Trace Formula for PSL(2,R), Vol. 2,Springer Lecture Notes,1001, Springer-Verlag, 1983.

  8. D. A. Hejhal, Eigenvalues for the Laplacian for Hecke triangle groups,Memoirs of AMS, Vol.469, 1992.

  9. D. A. Hejhal, D. Rackner, On the topography of Maass wave forms,Exper. Math.,1 (1992), 275–305.

    MATH  Google Scholar 

  10. J. Hoffstein, P. Lockhart, Coefficients of Maass forms and the Siegel zero, appendix by D. Goldfeld, J. Hoffstein, D. Lieman, An affective zero free region,Annals of Math.,140 (1994), 161–181.

    Article  MATH  Google Scholar 

  11. H. Iwaniec, Prime geodesic theorem,J. Reine Angew. Math.,349 (1984), 136–159.

    MATH  Google Scholar 

  12. H. Iwaniec, Small eigenvalues for Γ0(N),Acta Arith., LVI (1990), 65–82.

  13. H. Iwaniec, The spectral growth of automorphic L-functions,J. Reine Angew. Math.,428 (1992), 139–159.

    MATH  Google Scholar 

  14. H. Iwaniec, H. Sarnak, L norms of eigenfunctions of arithmetic surfaces, To appear inAnnals of Math.

  15. H. Iwaniec, Non-holomorphic modular forms and their applications,Modular forms, Durham conference, edited by R. Rankin, 1984, 157–196.

  16. D. Jakobson,Quantum ergodicity for Eisenstein series on PSL2(Z)\PSL2(R), Preprint, Princeton, 1994.

  17. L. Kuipers, H. Niederreiter,Uniform distribution of sequences, New York, Wiley, 1974.

    MATH  Google Scholar 

  18. N. V. Kuznetsov, Petersson’s conjecture for cusp forms of weight zero and Linnik’s conjecture, Sums of Kloosterman sums,Mat. Sb.,111 (1980), 334–383.

    Google Scholar 

  19. T. Meurman, On the order of the Maass L-function on the critical line,Number Theory, Vol. I, Budapest, 1987, Colloq. Math. Soc. Janos Bolyai,51 (1990), 325–354.

    Google Scholar 

  20. S. Ramanujan, Some formulae in the arithmetic theory of numbers,Messenger of Math.,45 (1916), 81–84.

    Google Scholar 

  21. B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces,Trans. AMS,233 (1977), 241–247.

    Article  Google Scholar 

  22. Z. Rudnick, P. Sarnak, The behavior of eigenstates of arithmetic hyperbolic manifolds,Com. Math. Phys.,161 (1994), 195–213.

    Article  MATH  Google Scholar 

  23. P. Sarnak,Arithmetic quantum chaos, The RA Blyth Lecture, University of Toronto, 1993.

  24. P. Sarnak,Some Applications of modular forms, Cambridge Univ. Press, 1990.

  25. A. Selberg,Collected Papers, Vol. 1, Springer-Verlag, 1989, 626–674.

    Google Scholar 

  26. A. I. Schnirelman, Ergodic properties of eigenfunctions,Usp. Math. Nauk.,29 (1974), 181–182.

    Google Scholar 

  27. G. Shimura, On the holomorphy of certain Dirichlet series,Proc. London Math. Soc. (3),31 (1975), 79–98,

    Article  MATH  Google Scholar 

  28. G. Steil, Über die Eigenwerte des Laplace Operators und die Hecke Operatoren für SL(2,Z), preprint, 1993.

  29. E. C. Titchmarsh,The Theory of the Riemann Zeta Function, Oxford, 1951.

  30. A. Weil, On some exponential sums,Proc. Nat. Acad. Sci. USA,34 (1948), 204–207.

    Article  MATH  Google Scholar 

  31. S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces,Duke Math. Jnl.,55 (1987), 919–941.

    Article  MATH  Google Scholar 

  32. S. Zelditch, Selberg trace formulae and equidistribution theorems,Memoirs of AMS, Vol. 96, No. 465, 1992.

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To Wolfgang Schmidt on the Occasion of His 60th Birthday

The research of the first author was supported by NSF Grant DMS 9304580, while he was a member at the Institute for Advanced Study during 1993–1994. The second author was partially supported by NSF Grant DMS 9102082.

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Luo, W., Sarnak, P. Quantum ergodicity of Eigenfunctions on PSL2(Z)/H 2 . Publications Mathématiques de l’Institut des Hautes Scientifiques 81, 207–237 (1995). https://doi.org/10.1007/BF02699377

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  • DOI: https://doi.org/10.1007/BF02699377

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