For context, imagine a sum along the variable $n$ whose terms $a_n = \lfloor nr\rfloor \cdot f(n)$ or $a_n = g(\lfloor nr\rfloor)$ for some value $r \in \mathbb{R}$. For example, I am working with the following sums (amongst others) with $r = \sqrt{2}$.
$$S=\sum_{n=1}^{\infty} \frac{\lfloor nr\rfloor}{n^3} \quad;\quad T=\sum_{n=1}^{\infty} \frac{H_{\lfloor nr\rfloor}}{n^3}$$
Where $H_k$ is the $k$-th harmonic number.
Is there an "easy" way to get rid of those $\lfloor nr\rfloor$?