## Nonvanishing of $$L$$-values and the Weyl law.(English)Zbl 1003.11019

Let $$\Gamma$$ be a Fuchsian group of the first kind, and let $$N_{\Gamma }(T)$$ be the counting function (in terms of $$t$$) for the eigenvalues $$1/4+t^2$$ of the hyperbolic Laplacian of the surface $$\Gamma \backslash \mathbb{H}$$, where $$\mathbb{H}$$ is the upper half-plane. The Weyl law is the claim $N_{\Gamma }(T) \sim (4\pi)^{-1}\mu (\Gamma \backslash \mathbb{H})T^2,$ where $$\mu$$ denotes the hyperbolic area. It is shown that this is false for generic hyperbolic surfaces under the assumption that the eigenvalue multiplicities of the Laplacian on $$\Gamma _0(p)\backslash \mathbb{H}$$, for a prime $$p$$, are bounded. Owing to the work of R. Phillips and P. Sarnak [Invent. Math. 80, 339-364 (1985; Zbl 0558.10017)], this assertion can be reduced to non-vanishing of “critical” values of certain Rankin-Selberg zeta-functions. In a previous paper [Duke Math. J. 69, 411-425 (1993; Zbl 0789.11032)], the author had proved such a result, which is however slightly too weak for the present purpose. This is now sharpened so as to show the non-vanishing property in a positive proportion of cases.

### MSC:

 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11M41 Other Dirichlet series and zeta functions

### Keywords:

Rankin-Selberg $$L$$-functions; Weyl law

### Citations:

Zbl 0558.10017; Zbl 0789.11032
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