Hardy-Littlewood Tauberian Theorem

Let $a_n\geq 0$ and suppose

$$\sum_{n=1}^\infty a_ne^{-an}\sim{1\over a}$$

as $a\to 0^+$. Then

$$\sum_{n\leq x} a_n\sim x$$

as $x\to\infty$.

See also Tauberian Theorem

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 118-119, 1994.