Mertens conjecture - bounds

Question

The disproven Mertens Conjecture states that $$|M(n)|\leq \sqrt{n}$$ If it is bounded at all, would the bounds $$|M(n)|\leq \sqrt{2n\log(\log (n))}$$ not be more realistic, and still consistent with the counterexamples suggested by Odlyzko and te Riele, owing to the law of the iterated logarithm?

Answer 1

We have that $\mid M(x)\mid\le x$ since the absolute value of the Moebius function is bounded by $1$. So it is indeed bounded. However, already a bound of the form, say, $O(x^{0.9})$ is not proven so far. This would of course follow from the Riemann Hypothesis. RH would imply a bound of the form $O(x^{1/2+\epsilon})$. For good unconditional bounds see the paper of Nathan Ng. Furthermore, a discussion with more information can be found here: Is $M(x)=O(x^σ)$ possible with $σ≤1$ even if the Riemann hypothesis is false?.