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For questions concerning limit superior and limit inferior of sequences (of real numbers and various generalizations) and also for $\limsup$ and $\liminf$ of sequences of sets.

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Behavior of $m(x)\sqrt{x}$ where $m(x)=\sum_{n\leq x}\frac{\mu(n)}{n}$

This is true by Exercise 4(b) for Section 15.1.1 in Montgomery-Vaughan: Multiplicative number theory I (CUP, 2006). The inequality contained therein can surely be proved in much the same way as (15.13 …
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