This is an extension of Emerton's comment. From my paperback 1990 reprint of "The Distribution of Prime Numbers" by A.E. Ingham (1932). Ingham is discussing your formula with the Mobius function, pages 105-106:
"Considerable importance was attached formerly to a function suggested by Riemann as an approximation to $\pi(x)$...This function represents $\pi(x)$ with astonishing accuracy for all values of $x$ for which $\pi(x)$ has been calculated, but we now see that its superiority over $\mathrm{Li}(x)$ is illusory...and for special values of $x$ (as large as we please) the one approximation will deviate as widely as the other from the true value."
In the next paragraph he shows that RH implies "And we can see in the same way that the function $\mathrm{Li}(x) - \frac{1}{2} \mathrm{Li}(x^{\frac{1}{2}})$ is `on the average' a better approximation than $\mathrm{Li}(x)$ to $\pi(x)$; but no importance can be attached to the latter terms in Riemann's formula even by repeated averaging."
Similar material is in Edwards pages 34-36 where he discusses relative size of terms, as Emerton comments. Edwards does not bother with comparing different approximations under the assumption of RH.