Short-arc equidistribution conjecture for Artin–Schreier L-functions

Let $\Psi$ be the set of nontrivial additive characters of $\mathbb{F}_p$, let $\mathcal{F}_d$ be the family of polynomials under consideration, and let $L_{f,\psi}$ have normalized zeroes on the unit circle. Fix $\psi\in\Psi$ and $C>0$. For each natural number $d$, let $I_d$ be any arc of length $C/d$. Short-arc equidistribution conjecture. As $d\to\infty$, the average number of zeroes of $L_{f,\psi}$ in $I_d$, with $f$ chosen uniformly from $\mathcal{F}_d$, is

$$\frac{C}{2\pi}+o(1).$$

This is the local random-unitary prediction for the family; the paper presents it as a conjecture and develops only partial results for related smooth statistics.