Extreme values of derivatives of zeta and L-functions

It is proved that as $$, uniformly for all positive integers $$, we have where $$. Here, $$ is the Dickman function. We have $$ and $$ when $$, which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet L-functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for $$ and $$. Furthermore, when assuming the Granville-Soundararajan conjecture is true, we establish the following asymptotic formulas: where $$ is prime and $$ is given.