Is there any conventional name for the series $$\sum_{k=1}^{\infty}\frac{\lfloor k\sqrt{x}\rfloor}{k^{s}}$$ where $x$ is a rational number, $s\geqslant3$ is a positive integer and $\lfloor \rfloor$ denotes the floor function? Are there any known results regarding this type of series?
Furthermore, I am mainly interested in $x$ being expressible as $\frac{p_1p_2\dots p_\alpha}{q_1q_2\dots q_\beta}$, where $p_i$, $q_i$ are primes, $p_i\neq q_j$, $p_i\neq p_j$, $q_i\neq q_j$, but I'm not sure if it is of any significance here.
EDIT: I have found the identity $$\sum_{k=1}^{\infty}\frac{\{ k\sqrt x \}}{k^{s}}=\frac{1}{2}\zeta(s)+\frac{1}{\pi}\Im(\sum_{n=1}^\infty \frac{\ln(1-e^{2\pi n\sqrt{x}i})}{n^s})$$ where $\Im$ stands for the imaginary part and $\{ \}$ is the fractional part, but I am now stuck on evaluating the series in the bracket. Any help?