All Questions
3 questions
9
votes
3
answers
544
views
Product of a Finite Number of Matrices Related to Roots of Unity
Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...
4
votes
1
answer
266
views
Positive system of algebraic integers
Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
4
votes
0
answers
437
views
A sum of Ramanujan sums
I have the following question about Ramanujan sums.
(All vectors and matrices here will be understood to have integer entries.)
Let $X_q=\{ (x_1,...,x_R)|1\leq x_i\leq q\} $ and let, for any $R\...