Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields

Abstract

We prove an effective version of the Chebotarev density theorem, conditional on the quasi-Riemann hypothesis for the Dedekind zeta function. Drawing on recent breakthroughs of OpenAI and Cureton--Thorner, we make unconditional the Ellenberg--Venkatesh bounds on $\ell$-torsion in class groups of number fields and extend an averaged result of Lemke Oliver--Smith to arbitrary families of degree-$d$-fields. Further applications are given to the least prime splitting and to small generators in number fields.