Zero density for automorphic L -functions

Abstract

In this paper, zero density estimates for automorphic L -functions L ( s , π ) for G L n are deduced from a bound for an integral power moment of L ( s , π ) on the critical line Re ( s ) = 1 / 2 . In particular for the Riemann zeta function, classical zero density estimates are extended to short vertical strips. For g being a holomorphic or Maass eigenform for S L 2 ( Z ) , bounds for zero density for L ( s , g ) in short strips are proved, which extend Ivićʼs results on long strips. For a self-dual Hecke Maass eigenform f for S L 3 ( Z ) , estimates of zero density for L ( s , f ) in short and long strips are also proved. The proofs use a zero detecting argument, a large sieve inequality, a bound for an integral power moment of L ( 1 / 2 + i t , π ) , the Rankin–Selberg theory, and the Halász–Montgomery–Jutila method.