Nonvanishing of $L$-values and the Weyl law


This work is concerned with the validity of Weyl law for hyperbolic surfaces on the asymptotic counting of the Laplace eigenvalues. Following Phillips-Sarnak, we show that the Weyl law is false for generic hyperbolic surfaces under the standard multiplicity assumption by establishing that a positive proportion of certain critical values of Rankin-Selberg $L$-functions do not vanish.



Wenzhi Luo