I saw one form of Riemann Hypothesis, it says: $$ \lim ∑(μ(n))/n^σ $$ Converges for all σ > ½

Is this statement same as the order of Mertens function is less than square root of n ?


Yes, since $\frac{1}{\zeta(\sigma)} = \sum{\frac{\mu(n)}{n^\sigma}}$, this is equivalent to the more canonical statement of RH that $\zeta$ has no zeroes to the right of the critical line.

You also mention $M(x) = O(x^{\frac{1}{2}+\epsilon})$, you can use the Mellin transform to show this is also equivalent to RH.

  • 2
    $\begingroup$ RH is actually equivalent to $M(x)=O(x^{1/2+\epsilon})$, the form you cited is stronger. $\endgroup$ – Markus Shepherd Jan 8 '15 at 16:25
  • $\begingroup$ Is it true that $M(x)=o(x^{\frac{1}{2}+\epsilon})$ is equivalent to RH? $\endgroup$ – mkultra May 23 at 12:33

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