The asymptotic behavior over a small parameter of a series of large deviation probabilities weighted with the Dirichlet divisors function

Abstract

We obtain the precise asymptotics of the series $$\sum_{k=1}^\infty \frac {d_k}k \mathtt{P}(|S_k| \ge \varepsilon k)$$ as $\varepsilon \downarrow 0$ where $S_k$ are partial sums of independent identically distributed random variables attracted to a stable law of index $\alpha \gt 1$.