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It is well known the case for sums like: $$ \sum_{i=0}^{p^n -1}\zeta^{-ai}, $$ where zeta is a primitive $p^n$-rooth of $1$. But, is there a standard formula for sums like: $$ \sum_{i=0}^{p^n -1}i^N\zeta^{-ai} $$ where $N$ is a fixed integer? Do you know references? Thanks for suggestions!

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  • $\begingroup$ Look up the formula for $\sum_{i = 0}^{m-1} i^Nx^i$ and then set $m=p^n-1$ and $x=\zeta^{-a}$. $\endgroup$ – KCd Mar 12 at 10:35
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    $\begingroup$ Start with a finite geometric series $1+x+\ldots+x^{m-1}$ and repeatedly differentiate and multiply by $x$ to make $x^i$ have coefficients that are powers of $i$. $\endgroup$ – KCd Mar 12 at 10:40
  • $\begingroup$ @KCd I think to have found the formula for when $i^N $ is costant but the general formula seems quite complicated $\endgroup$ – andres Mar 13 at 9:41
  • $\begingroup$ I have no idea what "when $i^N$ is constant" means ($i$ is changing, so it's not constant), but in any case sure, the formula is quite complicated. There's no reason to expect tidy formulas for such things as $N$ grows. $\endgroup$ – KCd Mar 13 at 10:46
  • $\begingroup$ @KCd I mean in the case when $i^N =c$ it's a simpler case. In general i think that it is not really possibile give a formula. In fact i think that it does not exists formulas for $\sum_i i^N$ too. $\endgroup$ – andres Mar 13 at 11:06

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