# Dirichlet and logarithmic densities share their domains, what can be said similarly?

It is a classical result that if $X$ is a subset of the positive integers $\bf N$, then the limit $$\lim_{x \to \infty} \frac{1}{\ln x}\sum_{n \in X\cap [1,x]}\frac{1}{n}$$ exists if and only if also the following limit exists $$\lim_{s \to 1^+} \frac{\sum_{n \in X} {1/n^s}}{\sum_{n \in {\bf N}} {1/n^s}}.$$

Moreover, in such case, the two limits are equal. [A proof can be found, for example, in Tenenbaum, Introduction to analytic and probabilistic number theory, II.Ch7].

Are there similar results about equivalence of existence of limits in a similar way?