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Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis
The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann $\zeta$-function is directly related to the growth of the Mertens function $M(x) \,=\,\sum_{k=1}^x \mu(k)$, where $\mu(k)$ is the M\"{o}bius coefficient of the integer $k$: the RH is indeed true if the Mertens function …