Oscillations of a Mobius sum Define $ S(n) = \sum_{k=1}^n {\mu(k)\over k}$ .  It is known but nontrivial that $S(n)$ approaches zero as $n$ approaches infinity.  
Here we are interested in the sign of $S(n)$ .  Note that the sign is always well defined since $S(n)$ is never equal to zero .   (For any prime $p$ in the range $\frac{n}{2} < p < n$ , the summand $-1/p$ is the only one involving that particular prime and therefore cannot be cancelled by the remaining terms.  And there is always such a prime by Bertrand's Postulate.)
Already for small values of $n$ , there are several changes of sign and it can be shown that $S(n)$ continues to oscillate about the origin.
Question:  Is it true asymptotically that $S(n)$ is positive half the time and negative half the time?
Thanks
 A: Let
\[\mathcal{P}^+ = \left\{x \geq 1 : \sum_{n \leq x} \frac{\mu(n)}{n} > 0\right\}, \quad \mathcal{P}^- = \left\{x \geq 1 : \sum_{n \leq x} \frac{\mu(n)}{n} < 0\right\}.\]
I very much doubt that
\[\frac{1}{X} \mathrm{meas}\left(\mathcal{P}^+ \cap [1,X]\right), \quad \frac{1}{X} \mathrm{meas}\left(\mathcal{P}^- \cap [1,X]\right)\]
have limits as $X \to \infty$ (it may be possible to prove this).
The correct notion is that of a limiting logarithmic density: we consider
\[\frac{1}{\log X} \int_{\mathcal{P}^+ \cap [1,X]} \, \frac{dx}{x}, \quad \frac{1}{\log X} \int_{\mathcal{P}^- \cap [1,X]} \, \frac{dx}{x}.\]
Then one can show conditionally that these converge to $1/2$ as $X \to \infty$, which confirms your question; one needs to assume the Riemann hypothesis, the Linear Independence hypothesis, and the bound
\[\sum_{0 < \gamma < T} \frac{1}{|\zeta'(1/2 + i\gamma)|^2} \ll T^{3 - \sqrt{3} - \delta}\]
for some $\delta > 0$.
This is essentially contained in the work of Akbary and Ng by combining Corollary 1.6 and Theorem 1.9; see also my MO answer here (the key point is the fact that the measure $\nu$ is even about the origin).
