Growth of a series involving the floor function to a certain exponent.

Let $$\theta \geq 0$$ and consider the sum $$\sum_{n \leq x} \lfloor \frac{x}{n} \rfloor^{-\theta}.$$.

How do I find a constant $$c(\theta)$$ so that this sum equals $$c(\theta)x+O(1),$$ where the implicit constans inside of the $$O$$ sign are independent of $$\theta?$$
A hint was to consider the identity
$$\sum_{n \leq x} f(n) G(x/n) = \sum_{m \leq x} (G(m)-G(m-1))F(x/m)$$ where $$F$$ is the summatory function of an arithmetic function $$f$$ and $$G$$ the summatory function of some arithmetic function $$g.$$