Can I get a PhD in artificial intelligence with a pure mathematics background?
You have some of the elements of an artificial intelligence major. Your background in logic is pertinent. And your background in mathematical analysis and Riemannian geometry is at least interesting.

What's troubling about what I've seen of your background is your lack of computer science courses. Maybe that's not an issue, if you have the equivalent of a minor in computer science that wasn't featured in your question. Certainly, that would be an important "tool." And if you don't have it, you might consider going back to school for a five or six course minor or certificate program in CS.

The most interesting part of your question was, "I find the Artificial Intelligence field really fascinating." Why is that? And how does that connect with the rest of your background? Or are you starting from scratch, which would be difficult, though not impossible. In any event, these issues should be addressed in your statement of purpose.

Here's my experience in unfamiliar territory: In high school, I had four years of French, and none of Spanish (unless you count my one year of self-taught Spanish). In college, I bluffed my way into a second year Spanish class, and completed it successfully. Here's an example where my interest and natural aptitude compensated for my lack of formal training. Perhaps it will be the same for you and artificial intelligence.