Evaluation of sums with the Möbius function $\mu (n) $

Question

assuming RH true , for big 'x' how would the sums

$$ \sum_{n=1}^{x} \mu (n) f(n) $$

as $ x \to \infty $ ?

my guess, if all the zeros have real part $ 1/2 $ then

$$ \sum_{n=1}^{x} \mu (n) f(n)\sim \int_{1}^{x}\frac{dt}{\sqrt{x}}f(t)cos(at+b) $$

here 'a' and 'b' are real numbers