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?


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