Long-power trace bound conjecture for Artin–Schreier L-functions

Assume $(d,p)=1$. Let $\Psi$ be the set of nontrivial additive characters of $\mathbb{F}_p$, let $\mathcal{F}_d$ be the relevant family, and write

$$T_{f,\psi}^r=\sum_{i=1}^{d-1}\rho_{f,\psi,i}^r$$

for the $r$-th power sum of the normalized zeroes. Long-power trace bound conjecture. For every $\psi\in\Psi$ and every $\varepsilon>0$, the average over $f\in\mathcal{F}_d$ satisfies

$$\left\langle T_{f,\psi}^r\right\rangle_{f\in\mathcal{F}_d}=O_{\varepsilon}\left(q^{\varepsilon r+(1/p-1/2)d}\right)$$

as $d\to\infty$ and $r\ge d$, with $q$ allowed to vary with $d$. This expresses the predicted decay of high trace moments suggested by uniform zero distribution.