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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  $L$   ‐functions on the level of the Bondarenko–Seip bound.

A dichotomy for extreme values of zeta and Dirichlet L $L$ ‐functions