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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, $p\in N$

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May I suggest that you not use $[x]$ for fractional part when everyone else uses it for integer part? Standard is $\lbrace x\rbrace$. –  Gerry Myerson Jul 9 '10 at 23:24
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I think the best hint is that the sum is independent of K when gcd(K,N) = 1. –  François G. Dorais Jul 10 '10 at 1:23
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up vote 3 down vote accepted

The article On Certain Sums of Fractional Parts 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.

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