A metric approach to zero-free regions for L-functions

For integers m,m' 1 m , m ′ ⩾ 1 , let π π and π ' π ′ be cuspidal automorphic representations of GL(m) GL ( m ) and GL(m') GL ( m ′ ) , respectively. We present a new proof of zero-free regions for L(s, π ) L ( s , π ) and for L(s, π 1.111pt× 1.111ptπ ') L ( s , π × π ′ ) under the assumption that π , π ' π , π ′ or L(s,π 1.111pt× 1.111ptπ ') L ( s , π × π ′ ) is self-dual. Our approach builds on ideas of “pretentious” multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic L -functions due to Lichtman and Pascadi.