Precise Tail Behaviour of Some Dirichlet Series

Let \(\eta _1\), \(\eta _2,\ldots \) 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 \(x\rightarrow 0+\), we derive precise tail asymptotics of the random Dirichlet series \(\sum _{k\ge 1}k^{-\alpha }\eta _k\) for \(\alpha \in (1/2, 1]\).