TITLE: What's the deal with Möbius pseudorandomness?
QUESTION [6 upvotes]: As I do more number theory, and in particular analytic number theory, I keep hearing more about the Möbius function $\mu(n)$ and how it is supposedly "pseudorandom". The values of the Möbius function at $n$ are determined by a formula, so what does this mean? Also, why is it so important that the Möbius function behaves pseudorandomly?
Just to be entirely clear, I am defining $\mu(n)$ by
\begin{equation}
\mu(n)=
\begin{cases}
(-1)^{\omega(n)}& n \,\,\mathrm{squarefree}\\
0&\mathrm{otherwise}
\end{cases}
\end{equation}
CLARIFICATION: This question was made with the intention that I would answer it myself. Any criticism, commentary, or discussion is appreciated.

REPLY [2 votes]: Here are a few (maybe helpful?) tidbits. In papers of 1931 and 1964 in the Comptes Rendus, Denjoy surmised that RH is true if the Mobius function was random for then your condition at (2) would be met. Also: true randomness would probably mean that the sequence needs to pass any statistical test that would be passed with probability one if the sequence consisted of statistically mutually independent random variables. In this latter case the order term in your condition (2) could actually be replaced by the square root of (x log log x) since the Law of the Iterated Logarithm would apply to it. There is also a time series character to this issue and for this recent work by Matomaki, Radziwill, Tao, Sarnak and others is relevant. Much remains unknown.