The present paper is concerned with Ω-estimates of the quantity (1/H)∫T+HT|(dm/dsm)ζk(12+it)|dt
where k is a positive number (not necessarily an integer), m a nonnegative integer, and (logT)δ≤H≤T, where δ is a small positive constant. The main theorems are stated for Dirichlet series satisfying certain conditions and the corollaries concerning the zeta function illustrate quite well the scope and interest of the results. %It is proved that if 2k≥1 and T≥T0(δ), then (1/H)∫T+HT|ζ(12+it)|2kdt>(logH)k2(loglogH)−C