21
votes
How small can a sum of a few roots of unity be?
This question grabbed my attention a couple of years ago and I've just put a paper on the arXiv with new upper bounds for $k=5$. I began by computing lots of data, then teased out the structure of ...
- 4,589
15
votes
Accepted
Vanishing of a sum of roots of unity
For general $N$, we can reason by induction on the $2$-adic valuation of $N$. If $N$ is odd, GH from MO's answer shows that $S_N :=\sum_{k=0}^{N-1} \zeta_{2N}^{2k^2+k} \neq 0$, where $\zeta_{2N} = z = ...
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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
11
votes
An algebraic number is not a root of unity?
Just noticed this question.
I think the following is an even more elementary/self-contained proof.
First, let $\eta = \xi u$ (so $u$ is a root of unity iff $\eta$ is),
and divide by $\xi^2$ to get
$$
...
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10
votes
A conjecture on binomial coefficients and roots of unity
Here is an elementary and explicit way to see this:
Suppose we have a set of $p$ integers $A=\{a_1, a_2,\dots, a_p\}$ which forms a complete set of residues modulo p. Then we have
$$\prod_{a\in A}(x-a)...
- 85.5k
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
8
votes
$q$ as a prime power and a root of unity
This is very not rigorous, but it's a way of thinking about this topic which I find personally helpful. Several $\mathbb F_1$ papers contains remarks about the idea going back to Weil and Iwasawa that ...
- 85.5k
6
votes
Summation formulas involving roots of unity to various powers
Experiment makes it clear that this sum is $\epsilon\delta\sqrt{p}$ where
$\epsilon=1$ if $p=1\pmod{4}$, and $\epsilon=i$ if $p=-1\pmod{4}$
$\delta=1$ if $\beta$ is a quadratic residue mod $p$, and $\...
- 56.9k
5
votes
Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture
Fourier transform does it.
Denote by $u_j$ $(j=0,1,\ldots,n-1$) the column-vector with coordinates $(x_{ji})_{1\leqslant i\leqslant n-1}$. Note that $u_0+u_1+\ldots+u_{n-1}=0$ and any $n-1$ vectors $...
- 109k
5
votes
$q$ as a prime power and a root of unity
One explicit connection is to look in non-describing characteristic, i.e. over an algebraically closed field $k$ of characteristic $p$ not dividing q, so $q$ can be simultaneously a root of unity and ...
- 2,242
5
votes
Accepted
Simplification of a sum with roots of unity
For fixed $k$, the product of $(1+\zeta^{jk}x)$ over $p$ consecutive values of $j$ equals $1+x^p$. Your claim follows.
- 109k
4
votes
How small can a sum of a few roots of unity be?
This is not an answer to the question, but I think it is somewhat related, and while the question deals with cyclic groups, this result deals with generalized characters of arbitrary finite groups. ...
- 44.4k
4
votes
Accepted
4
votes
Accepted
Norm of $2^{i}$-th primitive root
Let $K=\mathbb{Q}[\sqrt{-2}]$. Then $\frac{1}{2}\sqrt{-2}(1-i)$ is an eighth root of unity in $L$ whose norm is $\frac{1}{2}\sqrt{-2}(1-i)\frac{1}{2}\sqrt{-2}(1+i)=\frac{1}{4}(-2)(1-i^2)=-1\in K$. [I'...
- 16.2k
3
votes
Accepted
A similar relationship between the generic cubic and the Lehmer quintic?
The map $s(r) = \frac{n+2 + nr - r^2}{1 + (n+2)r}$ cyclically permutes the roots. This map is given in [2], and I found it through the reference in [1]. It turns out to give the correct order.
...
- 7,226
3
votes
A determinant involving the cotangent function
The conjecture has been proved by my graduate student Han Wang and me via the Eigenvector-eigenvalue Identity, for our joint paper see http://arxiv.org/abs/2206.02589 .
- 15.6k
3
votes
A conjecture involving roots of unity
Motivated by Nemo's solution in the case $\delta=0$, here I provide a proof for the case $\delta=1$. Let $\zeta$ be a primitive $m(n-1)+1$-th root of unity, and consider
$$S=\sum_{k=1}^{n-1}\left(\...
- 15.6k
3
votes
Trace 0 and Norm 1 elements in finite fields
No, not for every $q,\ell,\zeta$. In fact in general there does not even exist one solution.
Trivial example: $\zeta \in \mathbb F_q^\times$ implies $\zeta^{1- q^i} = 1$ and hence $Tr( \zeta^{1-q^i})=...
- 148k
2
votes
How to prove an approximation of a combinatorics identity
You could show that it is one of $r$ approximately equal things which add to $2^n.$ You might also bound how far apart they can be from each other. This is really pretty much what Fedor said but the ...
- 30.1k
2
votes
A conjecture involving roots of unity
This proves the case $\delta=0$, namely for $z^{mn-1}=1$
$$
S=\sum_{k=1}^{n-1}\left(\frac{z^k}{1+z^{km}}-(-1)^{n-k}\frac{z^k}{1-z^{km}}\right),\quad \text{Re} \,S=(-1)^{n-1}\left\lfloor \frac n2\right\...
- 5,624
2
votes
Accepted
Roots of anti-palindromic polynomial if coefficients are odd.
$P$ has no roots in $\mathbb{U}$. Ad absurdum, assume that there exists $P \in \mathbb{Z}[X]$ with degree $n \ge 1$, such that :
$P(-X) = X^nP\Big(\frac{1}{X}\Big)$,
all the coefficients of $P$ from ...
- 188
2
votes
Möbius inversion formula and roots of unity
In case of the root of unity is fixed one for all,
From the orthogonality of Dirichlet characters you can decompose the periodic sequence $n\to \zeta_{p^k}^n $ as a linear combination of the $\chi(\...
- 3,403
2
votes
Third roots of unity and norm element
Of course you can check whether $\zeta_3$ is the norm of a unit by computing the unit group. There are a few publications about Scholz's unit knot (see Jehne's article https://eudml.org/doc/152174 ).
...
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1
vote
Points on a circle with near-zero centroid
Let me try to point you in the right direction. Let $M_N$ denote the quotient of the $N$-fold product
$$
S^1\times ... \times S^1
$$
by the action of the group of rotation $SO(2)=S^1$:
$$
(z_1,...,z_N)...
- 31.2k
1
vote
A conjectural permanent identity
The two conjectures in the posting and postscript (concerning (3) and (5)) have been confirmed in the following paper via a novel permanent identity.
Yue-Feng She, Zhi-Wei Sun and Wei Xia, A novel ...
- 15.6k
1
vote
Identities involving derangements and roots of unity
My idea is to define a $(n-1)\times(n-1)$ Hermitian matrix with diagonal elements equal to zero and off-diagonal elements $a_{ij}= 1/(1-x_{i-j})$, where $x_k=\zeta^k$. So the left-hand side of (4) is ...
1
vote
Accepted
Q-binomials at roots of unity
So, the canonical answer is the q-Lucas theorem, as pointed out in the comments.
This was proved in
Olive, Gloria, Generalized powers, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.
- 15.8k
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