Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of this vector is $|\mathcal{F}_n| = 1 + \sum_{m=1}^n \varphi(m). $ Let's call $k$ a root when $\vec v_k \cdot \vec v_0=0$.

I tried for quite a while to get a set $OV$ of $v_k$, where $k$ can be $0$, positive or negative, such that the collection of vectors $\vec v_k$ creates a unitary matrix. The largest sets I found, have six elements, e.g. $n=40$: $OV=\{0, 1,-179,180,-29748,29749 \}$.

I also found that the roots for even $n$ fall into certain categories. For $n=40$ we get:

  • $2^23^25p$, e.g. $\color{red}{2^23\cdot 5=180}$
  • $2^23\cdot 37^np$, e.g. $\color{blue}{2^23\cdot 37\cdot 67=29748}$
  • $2^33\cdot 29^np$, e.g. $\color{green}{2^33\cdot 29\cdot 43=29928}$
  • $2^43\cdot 29^np$
  • $2^33\cdot 7^k23^np$,

where $n,k>0$ and $p$ may be any prime larger than $40$. Roots of odd $n$ are products of primes larger than $40$.

While analyzing this special set for $n=40$, I found that at least, either the sum or the difference of any pair would give rise to another root:

  • $\vec v_0 \cdot \vec v_{(-\color{blue}{29748}-\color{red}{180})}=\vec v_0 \cdot \vec v_{-\color{green}{29928}}=0$, whereas $\vec v_0 \cdot \vec v_{(-29748+180)}\neq 0$.
  • $\vec v_0 \cdot \vec v_{(29749\pm 180)}=0$, where $29749\pm 180$ is either prime or a product of primes larger than $40$.

There seems to be kind of "closed under addition/subtraction"-relation among the roots. Is this an equivalence class?

My questions:

Are these matrices already known? How/where are they used? How to create them? What about these closedness relations?


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