A generalization of Möbius inversion formula guarantees that if we have $G(n)=\sum_{k=1}^{n} F\left(\frac{n}{k}\right)$, then $F(n)=\sum_{k=1}^{n} \mu(k) G\left(\frac{n}{k}\right)$.
If we have $\sqrt{n}<x<n$, could it be correct if we state that if we have $G(n)=\sum_{k=1}^{x} F\left(\frac{n}{k}\right)$, then $F(n)=\sum_{k=1}^{x} \mu(k) G\left(\frac{n}{k}\right)$?
In case the above statement is generally incorrect, could it be correct depending on $F(n)$ and $G(n)$?