Zeros of Families of Automorphic L-functions

For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square L-functions of classical automorphic forms. For analytic purposes, in the absence of the Generalized Riemann Hypothesis, it is very useful to have an upper-bound, on average, for the number of zeros of the L-functions which are very close to 1. We prove a very general statement of this type for forms on GL(n)/Q for any n 1, comparable to the log-free density theorems for Dirichlet characters first proved by Linnik.