Riemann Hypothesis, is this statement equivalent to Mertens function statement?

Question

All:

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 ?

Answer 1

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.