Farey Sequence Vector Orthogonality Relation Question

Question

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. $\vec v_0$. For all $k\in OV=\{ 1, 17 ,19 ,28 ,41 ,43 ,47 ,53 ,59 ,61 ,67,...\}$ I get vectors orthogonal to $\vec v_0$. The integer sequence $OV$ doesn't seem to be know to OEIS. As Greg pointed out the set is periodic with period $\mathop{\rm lcm}[1,\dots,39] = 5342931457063200$.

$OV$ contains all primes larger than $39$ (denoted $p_{\gt39}$) and all products of them.

All numbers $k\pmod 2 \equiv 0$, I found$^*$ are either of the form $2^27^np_{\gt 39}$ or $2\cdot 3^27^np_{\gt39}$.

All numbers $k\pmod 5 \equiv 0$, I found$^*$ are either of the form $5\cdot 7 \cdot 11 p_{\gt 39}$ or $5\cdot 13\cdot 23p_{\gt39}$.

$17$ and $19$ complete the list of primes till $23$, leaving a gap containing $29,31$ and $37$.

How can the products of small primes and the gap be explained?

$^*: {\scriptstyle\text{I searched the first $122827$ roots...}}$

Answer 1

This isn't a complete answer but rather a way to start propertly investigating the problem. The sum of $\exp(2\pi i k b/q)$ over all reduced fractions $b/q$ with denominator $q$ is equal to Ramanujan's sum $c_q(k)$. So the quantity you're describing is $$ \vec v_0\cdot\vec v_k = \sum_{a_m\in\mathcal F_{39}} \exp(2\pi i k a_m) = \sum_{j=1}^{39} c_j(k). $$ For example, when $k=1$, it's known that $c_j(k) = \mu(j)$ (the Möbius mu function). And sure enough, $\sum_{j=1}^{39} \mu(j)=0$.

A formula for $c_j(k)$ is known for all $j$ and $k$, so you can start looking at that. One consequence is that your set $OV$ is actually periodic: it's a union of residue classes modulo $\mathop{\rm lcm}[1,\dots,39] = 5342931457063200$.