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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?

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    [Sorry - in the following comment I read the requirement on $S$ to be asymptotic density $1/2$ or above...] With respect to your second question I think the answer is no. For instance you would take $S=2\mathbb{N}$ - a set which will have asymptotic density $1/2$ - in which case the series will converge.2012-06-29
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    Two ways of arranging the punctuation: $\begin{cases} A & \text{if }B \\ C & \text{if }D \end{cases}.$ ${}\qquad{}$ $\begin{cases} A & \text{if }B, \\ C & \text{if }D. \end{cases}$ ${}\quad{}$ I changed the first to the second.2012-06-29
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    I made the other question into an other question http://math.stackexchange.com/questions/164655/sum-sum-n1-gn-n-over-a-set-of-with-density-1-22012-06-29

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Let $f(n)=-1$ if $n$ is prime, and $f(n)=1$ otherwise. We show that the sum $$\sum_{n=6}^\infty \frac{f(n)}{n}$$ does not converge.

Arrange the integers $\ge 6$ in consecutive groups of $6$. In the set $\{6k,6k+1,6k+2,6k+3,6k+4,6k+5\}$, the numbers $6k+1$ and $6k+5$ may be prime. The other four are definitely composite. It follows that $$\sum_{i=0}^5 \frac{f(6k+i)}{6k+i} \ge \frac{1}{6k}-\frac{1}{6k+1}+\frac{1}{6k+2}+\frac{1}{6k+3}+\frac{1}{6k+4}-\frac{1}{6k+5}.$$

The sum on the right is $\gt \frac{1}{6k+2}+\frac{1}{6k+3}$, which in turn is $\gt \frac{2}{6k+3}$.

But $\sum_{k=1}^\infty \frac{2}{6k+3}$ diverges. So the sequence of partial sums of the shape $\sum_{n=6}^{6k+5} \frac{f(n)}{n}$ is unbounded, and therefore the series of the question does not converge.

The argument proves a little more: the series in fact diverges to $\infty$.

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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 $$S(n) = \gamma - 2B + \log \left( \dfrac{n}{\log^2 (n)} \right) + \mathcal{O} \left( \dfrac1{\log n}\right)$$