Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten’s function for $n=39$ is zero $$ M(39)= \sum_{a\in \mathcal{F}_{39}} e^{2\pi i a} =0 , $$ our vector $\vec v_1$ is orthogonal to the vector containing only $1$’s, i.e. $\vec […]

A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$. For example $F_6= \{0,1/6,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,5/6,1\}$. The consecutive differences of $F_6$ are $S_6=\{1/6, 1/30, 1/20, 1/12, 1/15, 1/10, 1/10, 1/15, 1/12, 1/20, 1/30, 1/6\}$, and the sum of squares of […]

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 […]

An (approximate) non-negative rational number representation is a pair of natural numbers each not greater than some fixed limit M (and of course denominator being non-zero). With this condition there is finite number of representable rational numbers. This means that for each such number we can name previous and next number in the set (of […]

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