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 mathrmRe(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.
