Natural density of a subset $S $ of positive integers is defined as $$\lim_{n \to \infty}\frac{1}{n}\sum\limits_{k\in S, k\le n}1 $$ whenever it exists. The logarithmic density of $S$ is defined as $$\lim_{n \to \infty}\frac{1}{\log n}\sum\limits_{k\in S, k\le n}\frac{1}{k} $$ if it exists.
It is known that even though the natural density does not exist, logarithmic density does. So, my question is why the logarithmic density is studied and what is its importance?
$$\sum_{n \in S} n^{-s} = \sum_n (\sum_{k \in S, k \le n} 1/k) (n^{1-s}-(n+1)^{1-s})$$ If $$\sum_{k \in S, k \le n} 1/k = (C+o(1))\sum_{k \le n} 1/k$$ then as $s \to 1^+$ $$\sum_{n \in S} n^{-s} = \sum_n ((C+o(1))\sum_{k \le n} 1/k)(n^{1-s}-(n+1)^{1-s})= (C+o(1)) \zeta(s)$$