The automaticity of the set of primes

The automaticity $A(x)$ of a set $\mathcal{X}$ is the size of the smallest automaton that recognizes $\mathcal{X}$ on all words of length $\le x$. We show that the automaticity of the set of primes is at least $x\exp (-c(\log \log x{)}^2\log \log \log x)$, which is fairly close to the maximal automaticity.