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I know that the following statement: For every $\epsilon>0$ $$M(N)=O(N^{0.5+\epsilon})$$ is equivalent to Riemann Hypothesis (Where $M(N)$ is Mertens function).

As Mertens function behaves somehow randomly here is idea to treat this function as a translation in random walk. Each step is related to values of independent random variable which is equal to $$-1,0,+1$$ with probability respectively equal to $$\frac{3}{\pi^{2}},1-\frac{6}{\pi^{2}},\frac{3}{\pi^{2}}$$ I know that the behaviour of translation will be more generally but on the other hand i saw a statement that given values of probability"uniquely determine the asymptotic behavior of the Mertens function"

I quoted it from this:

https://arxiv.org/ftp/arxiv/papers/1712/1712.04674.pdf

Does have it sense to use here probabilistic tools such as central limit theorem or law of iterated logarithm?

Thanks in advance for answers.

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