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

Abstract For integers $$m,m' \geqslant 1$$ m , m ′ ⩾ 1 , let $$\pi $$ π and $$\pi '$$ π ′ be cuspidal automorphic representations of $$\textrm{GL}(m)$$ GL ( m ) and $$\textrm{GL}(m')$$ GL ( m ′ ) , respectively. We present a new proof of zero-free regions for $$L(s, \pi )$$ L ( s , π ) and for $$L(s, \pi \hspace{1.111pt}{\times }\hspace{1.111pt}\pi ')$$ L ( s , π × π ′ ) under the assumption that $$\pi , \pi '$$ π , π ′ or $$L(s,\pi \hspace{1.111pt}{\times }\hspace{1.111pt}\pi ')$$ 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.