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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 $g.$

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