Find the sum if possible and the asymptotic expansion to high order of $$W \left({m, n}\right) = \sum_{k=1}^{\lfloor{m/2}\rfloor} \left\{{\sqrt{k^2+n}}\right\}$$
where $\left\{{...}\right\}$ is the fractional part. Also $m < n$ applies to the problem that I am working on but this condition can be relaxed for this solution. For the first approximation assume that $\sqrt{k^2+n}$ is equally distributed then $\left\{{x}\right\} = 1/2$ or $$W \left({m, n}\right) \sim \sum_{k=1}^{\lfloor{m/2}\rfloor} \frac{1}{2} \sim \frac{1}{4} m$$ Numerical testing shows that $$W \left({m, n}\right) \sim \left[{0.21-0.24}\right] m$$ with a weak dependence $n$. This has been tested up to $m=10^7$ with various values of $n$ such as $n = m$, $n=10m$, and $n=100m$. A good asymptotic expansion is needed because the full asymptotic expansion of the other terms of my problem gives $(1/2)m\left({m+2}\right)$. So the approximation above will cause larger errors as $m \rightarrow \infty$.