A dichotomy for extreme values of zeta and Dirichlet L-functions

TL;DR

This paper demonstrates large values of the Dedekind zeta function in cyclotomic fields, revealing a dichotomy between improving bounds for the Riemann zeta function or for Dirichlet L-functions.

Contribution

It establishes a new connection between large values of Dedekind zeta functions and bounds for Riemann zeta and Dirichlet L-functions, highlighting a fundamental dichotomy.

Findings

Large values of Dedekind zeta functions on the critical line

Implication of a dichotomy in bounds for zeta and L-functions

Potential pathways for improving bounds in analytic number theory

Abstract

We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of Dirichlet $L$-functions on the level of the Bondarenko--Seip bound.