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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9$\begingroup$ May I suggest that you not use $[x]$ for fractional part when everyone else uses it for integer part? Standard is $\lbrace x\rbrace$. $\endgroup$– Gerry MyersonJul 9, 2010 at 23:24
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2$\begingroup$ I think the best hint is that the sum is independent of K when gcd(K,N) = 1. $\endgroup$– François G. DoraisJul 10, 2010 at 1:23
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1 Answer
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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.