Precise tail behavior of some Dirichlet series

Let ${\eta }_1$, ${\eta }_2,\dots$ be independent copies of a random variable $\eta$ with zero mean and finite variance which is bounded from the right, that is, $\eta \le b$ almost surely for some $b>0$. Considering different types of the asymptotic behaviour of the probability $\mathbb{P}\{\eta \in [b-x,b]\}$ as $x\to 0+$ we derive precise tail asymptotics of the random Dirichlet series ${\sum }_{k\ge 1}{k}^{-\alpha }{\eta }_k$ for $\alpha \in (1\mathrm{/}2,1]$.