  Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth,
projective, geometrically connected curves over a perfect field of
characteristic $p>0$. We investigate the relationship between the $a$-numbers
of $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to the
relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the
Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special
situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$
and the ramification of the cover, so we instead give bounds on the $a$-number
of $Y$. We provide examples showing our bounds are sharp. The bounds come from
a detailed analysis of the kernel of the Cartier operator.
