let $x/\mathbb{f}_q$ be a smooth affine curve over a finite field of characteristic $p > 2$. in this talk we discuss the $p$-adic variation of zeta functions $z(x_n,s)$ in a pro-covering $x_\infty:\cdots \to x_1 \to x_0 = x$ with total galois group $\mathbb{z}_p$. for certain ``monodromy stable'' coverings over an ordinary curve $x$, we prove that the $q$-adic newton slopes of $z(x_n,s)/z(x,s)$ approach a uniform distribution in the interval $[0,1]$, confirming a conjecture of daqing wan. we also prove a ``riemann hypothesis'' for a family of galois representations associated to $x_\infty/x$, analogous to the riemann hypothesis for equicharacteristic $l$-series as posed by david goss. this is joint work with joe kramer-miller.