Any hints how to compute this sum
$$\sum\_{i=1}^{N-1}\left[i\frac{K}{N}\right]^{p}?$$
where K < N , $\left[\cdot\right]$ denotes fractional part,
$p\in N$



sum of  fractional parts
The article [On Certain Sums of Fractional Parts](https://doi.org/10.1007/BF01238638 "Arch. Math 25, 41–44 (1974). zbMATH review at https://zbmath.org/?q=an:0277.10005") by Gandhi and Williams answers your question for $p=1$; it's likely that since 1974 this result has been generalized, but I wasn't able to find a reference.


