Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e.
$$x \equiv x\ \%\ m\pmod{m}$$
Let $\mu^n_m(x)$ denote multiplication by $n$ modulo $m$, i.e.
$$\mu^n_m(x) = nx\ \%\ m$$
Consider the Fourier transform of $\mu^n_m(x)$, i.e.
$$\tilde{\mu}^n_m( k) = \frac{1}{m}\sum_{x=0}^{m-1}e^{i2\pi k x/m}\mu^n_m(x)$$
From user GH from MO's answer to another MO-question I've learned that
$$\tilde{\mu}^n_m(k) = -\frac{1}{2}-\frac{i}{2}\cot\frac{\pi \bar nk}{m}$$
with $\bar n$ the multiplicative inverse of $n$ modulo $m$.
A suggested analogy between the Fourier transform $\tilde{\mu}(k)$ and Riemann's hypothesis seemed too far-fetched, and there's only little evidence that there is a deeper connection:
"The one has to do with coprimes, the other with primes."
"Both exhibit some irregularity and quasi-randomness."
"In both $\operatorname{Re}(x) = \pm 1/2$ plays a somehow similar role."
But I wonder if a deeper connection between the Fourier transform $\tilde{\mu}(k)$ and the roots $\zeta(k)$ of polynomials – especially their Galois groups – may be motivated by the following analogies.
Let $\zeta_m(k)$ be the complex roots (indexed by $k$) of a monic polynomial $P_m(x) = x^m + \sum_{k=0}^{m-1}a_kx^k$ with coefficients from $[m] = \{0,1,\dots,m-1\}$.
$$\begin{array}{r|l|l|} & \tilde{\mu}^n_m & \zeta_m \\ \hline \text{as functions} & [m] \rightarrow \mathbb{C} & [m] \rightarrow \mathbb{C} \\ \hline \text{sum} & \sum_{k=0}^{m-1} \tilde{\mu}^n_m( k) = 0 & \sum_{k=0}^{m-1} \zeta_m(k) = -a_{m-1} \\ \hline \text{conjugate pairs} & z \in \tilde{\mu}^n_m \equiv \overline{z} \in \tilde{\mu}^n_m & z \in \zeta_m \equiv \overline{z} \in \zeta_m \\ \hline \text{permutation symmetry among} & \operatorname{Re}(\tilde{\mu}^{n_1}_m) = \operatorname{Re}(\tilde{\mu}^{n_2}_m) & \text{?} \\ \hline \text{possible multiplicities for} & \tilde{\mu}^n_m(k) = 0\ & \text{??} \\ \hline \text{closed form} & \text{depends on } \overline{n} & \text{depends on Galois group} \\ \hline \end{array}$$
There's one major difference: The Fourier coefficients $\tilde{\mu}(k)$ are genuinely ordered – and $\tilde{\mu}(0)$ plays a special role – while the roots $\zeta(k)$ are not – and $\zeta(0)$ is not even well defined.
It might be revealing to especially consider monic polynomials
$P^n_m(x)$ with coefficients $a_k = \mu^n_m(k)$ for some $n,m$ – with roots $\zeta^n_m(k) \in \mathbb{C}$
$P^n_m(x) \equiv 0 \pmod{m}$ instead of $P^n_m(x) = 0$ – with roots $\zeta^n_m(k) \in [m]$
My questions are:
How can the gaps $?$ and $??$ in the table above be closed in a concise way?
What might knowing $\tilde{\mu}^n_m(k)$ tell us about $\zeta^n_m(k)$ and vice versa?
