Selberg's limit theorem for $L$-functions associated to $GL(n)$ Hecke-Maass cusp forms

The Selberg's limit theorem asserts that under an appropriate normalization, the logarithm of the Riemann zeta-function on the critical line is normally distributed. This investigation was extended to the imaginary part of the logarithm of the Dirichlet $L$-functions associated to a family of primitive characters, as well as the Hecke $L$-functions associated to a family of $GL(2)$ or $GL(3)$ Hecke-Maass cusp forms. We shall report a result on the case of $GL(n)$, where $n$ is at least $3$, using the automorphic Plancherel density theorem of Matz and Templier. This is a joint work with Guohua Chen and Yingnan Wang.

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