  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, $\eta\leq b$ 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
$\sum_{k\geq 1}k^{-\alpha}\eta_k$ for $\alpha\in (1/2, 1]$.
