Displaying similar documents to “Quantum ergodicity of Eigenfunctions on P S L 2 ( 𝐙 ) / 𝐇 2 ”

New integral representations for the square of the Riemann zeta-function

Andreas Guthmann (1997)

Acta Arithmetica

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Introduction. The recent discovery of an analogue of the Riemann-Siegel integral formula for Dirichlet series associated with cusp forms [2] naturally raises the question whether similar formulas might exist for other types of zeta functions. The proof of these formulas depends on the functional equation for the underlying Dirichlet series. In both cases, for ζ(s) and for the cusp form zeta functions, only a simple gamma factor is involved. The next simplest case arises when two such...

Quantum unique ergodicity for Eisenstein series on P S L 2 ( ℤ ∖ P S L 2 ( ℝ )

Dmitry Jakobson (1994)

Annales de l'institut Fourier

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In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on P S L 2 ( ℤ ) ∖ P S L 2 ( ℝ ) . This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for P S L 2 ( ℤ ) ∖ ℍ . The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of S L 2 ( ℝ ) . In the proof the key estimates come...

The reciprocal of the beta function and G L ( n , ℝ ) Whittaker functions

Eric Stade (1994)

Annales de l'institut Fourier

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In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself. We then apply our new formula to the study of G L ( n , ℝ ) Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral...

Prime geodesic theorem

Yingchun Cai (2002)

Journal de théorie des nombres de Bordeaux

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Let Γ denote the modular group P S L ( 2 , Z ) . In this paper it is proved that π Γ ( x ) = li x + O ( x 71 102 + ϵ ) , ϵ > 0 . The exponent 71 102 improves the exponent 7 10 obtained by W. Z. Luo and P. Sarnak.