Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis

  title={Randomness of M{\"o}bius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis},
  author={Giuseppe Mussardo and Andr{\'e} LeClair},
  journal={Journal of Statistical Mechanics: Theory and Experiment},
  • G. MussardoA. LeClair
  • Published 25 January 2021
  • Mathematics
  • Journal of Statistical Mechanics: Theory and Experiment
The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function M(x)=∑k=1xμ(k) , where μ(k) is the Möbius coefficient of the integer k; the RH is indeed true if the Mertens function goes asymptotically as M(x) ∼ x 1/2+ϵ , where ϵ is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global… 
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