Uniform-distribution conjecture for the random variable X_l(m,N)

In fact, for every odd integer $N\ge3$ and every $\ell$ coprime to $N$, the random variable $X_\ell(2,N)$ is uniformly distributed, regardless of whether $N$ is prime.

Put

$$\zeta=\exp\left(-\frac{2\pi i\ell}{N}\right).$$

Since $\gcd(\ell,N)=1$, its powers are precisely the $N$ distinct $N$-th roots of unity. The variable $X_\ell(2,N)$ chooses one of the

$$\binom N2$$

unordered pairs of distinct roots with equal probability.

No such pair has sum zero: otherwise its elements would be antipodal, which is impossible when $N$ is odd. Now let $z,w$ be any two unit-modulus roots with

$$s=z+w\ne0.$$

Because $\overline z=1/z$ and $\overline w=1/w$,

$$\overline s = \frac1z+\frac1w = \frac{s}{zw}.$$

Consequently,

$$zw=\frac{s}{\overline s}.$$

Thus the unordered pair $\{z,w\}$ is uniquely determined by its sum: its members are exactly the two roots of

$$T^2-sT+\frac{s}{\overline s}=0.$$

Therefore all $\binom N2$ pair sums are distinct, and each occurs with probability

$$\frac1{\binom N2}.$$

Taking

$$N=9,\qquad \ell=1,\qquad m=2$$

gives $36$ equally probable values although $N$ is composite. All stated hypotheses hold:

$$2\le m\le N-1,\qquad \gcd(\ell,N)=\gcd(m,N)=1.$$

More generally, every odd composite $N$ provides a counterexample.

The source's Proposition 2.1 already establishes uniformity at the allowed endpoint $m=N-1$ for every $N$. The interior family $m=2$ above shows that excluding this endpoint would still not repair the conjecture.