A spectral universality theorem for Maass $L$-functions

Abstract

We show that for a positive proportion of Laplace eigenvalues ${\lambda }_j$ the associated Hecke-Maass $L$-functions $L(s,u_j)$ approximate with arbitrary precision any target function $f(s)$ on a closed disc with center in $3/4$ and radius $r<1/4$. The main ingredients in the proof are the spectral large sieve of Deshouillers-Iwaniec and Sarnak's equidistribution theorem for Hecke eigenvalues.

12 pages, final version, corrected typos