Generalization of Selberg's 3/16 Theorem and Affine Sieve

摘要

A celebrated theorem of Selberg states that for congruence subgroups ofSL(2,Z) there are no exceptional eigenvalues below 3/16. We prove ageneralization of Selberg's theorem for infinite index "congruence" subgroupsof SL(2,Z). Consequently we obtain sharp upper bounds in the affine linearsieve, where in contrast to \cite{BGS} we use an archimedean norm to order theelements.