Precise Tail Behaviour of Some Dirichlet Series

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