TITLE: the sum $\sum \limits_{n>1} f(n)/n$ over primes
QUESTION [7 upvotes]: Let 
$$
f(n)=\begin{cases}-1&\text{if $n$ is a prime integer},\\
1&\text{otherwise}.
\end{cases}
$$ 
Then, does the series
$$
\sum_{n>1} f(n)/n
$$
converge or diverge?

REPLY [6 votes]: André Nicolas has already shown that the series diverges. The divergence can be quantified with relative ease as well.
$$ S(n) = \sum_{k=1}^{n} \dfrac{1 - 2 \times 1_{\text{$k$ is a prime}}}k = \sum_{k=1}^{n} \dfrac1n - 2 \sum_{k- \text{ prime }\leq n} \dfrac1k \\ = \left(H_n - \log n \right) - 2 \left(\sum_{k- \text{ prime }\leq n} \dfrac1k - \log (\log n) \right) + \log (n) - 2 \log \log (n)$$
From the asymptotic of $H_n$ and $\sum_{k- \text{ prime }\leq n} \dfrac1k$, we have that $$S(n) = \gamma - 2B + \log \left( \dfrac{n}{\log^2 (n)} \right) + \mathcal{O} \left( \dfrac1{\log n}\right)$$