Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) series $\sum_{n \ge 1} a_n e^{-\lambda_n s}$ converges for all $s > 0$, as its abscissa of convergence is $0$ by the main result of Chapter II, Section 6 from:

G. H. Hardy and M. Riesz,

The General Theory of Dirichlet's Series, Cambridge Univ. Press: Cambridge, 1915.

So my question is:

Q.What about sufficient conditions for having that $\sum_{n \ge 1} a_n e^{-\lambda_n s} \sim \frac{1}{s}$ as $s \to 0^+$?

*To be clear:* I'm just looking for references.

*For the record:* I'm especially, though not uniquely, interested in the case where $a_n = n^{\alpha}$ for all $n$, with $\alpha$ being a given exponent $\ge -1$.