Extreme values of derivatives of zeta and -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 -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.