Precise Tail Behaviour of Some Dirichlet Series

Let η 1 , η 2 , … be independent copies of a random variable η with zero mean and finite variance which is bounded from the right, that is, η ≤ b almost surely for some b > 0 . Considering different types of the asymptotic behaviour of the probability P { η ∈ [ b - x , b ] } as x → 0 + , we derive precise tail asymptotics of the random Dirichlet series ∑ k ≥ 1 k - α η k for α ∈ ( 1 / 2 , 1 ] .