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Results tagged with roots-of-unity
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user 11919
8
votes
Power sums of p-th roots of unity
I think I can answer my own question (which is really my colleague's question).
Let $p>2$, and let $j\in\{1,\dots,p-1\}$ be fixed. Then
$$ \sum_{k=1}^{p-1}\left|z_1^k+\dots+z_j^k\right|^2 = \sum_{k= …
- 105k
9
votes
Accepted
Summation formulas involving roots of unity to various powers
Your first sum is a special Gauss sum. For its value in general, see Corollary 9.16 in Montgomery-Vaughan: Multiplicative number theory I. Your second sum can also be expressed in terms of Gauss sums …
- 105k
3
votes
Non-standard Gauss sums
In the ring $\mathbb{Z}[\omega_p]$, the OP's second sum $\sum_{k=1}^{p-1} \left(\frac{k^2+k}{p}\right) \omega_p^{kl}$ raised to the $p$-th power is congruent to
$\sum_{k=1}^{p-2} \left(\frac{k^2+k}{p …
- 105k
12
votes
Vanishing of a sum of roots of unity
The sum does not vanish when $N$ is odd. I will elaborate on François Brunault's idea.
Assume that $N$ is odd, and let $S$ be the sum in question. Working in the ring $\mathbb{Z}[z]$, we see that $S^{ …
- 105k