I am currently finishing my master in pure mathematics. I've studied mainly logic, mathematical analysis a bit of probability and I'm writing my thesis about Riemannian Geometry. I've been thinking about doing a PhD, but I am not so sure if I really want (and if I am good enough) to stay in the Academia for all my life.
Even if I don't know anything about, I find the Artificial Intelligence field really fascinating but I know that my current background is inadequate.
Do you think I could get I PhD in that area (possibly using also my mathematical background)?
And what kind of career path could I follow after? I mean, is it easy to find a job related to AI?
I just want to supplement other answers (I have a PhD in AI). If your background is mathematics, I do not agree with others that you need to code per-se. Aside from machine learning pointed above, there are some other sides you can look at that do not require - much - programming (if any).
Logic - Symbolic AI is broadly focused around temporal logics, including those specifically formalised for game-theory, and non-monotonic logics. You can go in many directions with logic in AI: philosophical (i.e., thinking of how to capture certain types of common-sense reasoning, normative reasoning, etc.) or more solution-oriented (i.e., we need a logic to represent a specific problem X and then we will prove some general results about that logic). In the case that you work on logics to solve specific problems, you may need some Computer Science understanding (in particular, proving complexity for this kind of research is required for publishing in many top venues). In terms of the more philosophical aspects, you don't really need so much of a Comp. Sci. background, non-monotonic logics, for example, can be generalised in a graph-theoretical framework.
Game theory - similar, to the problem-oriented logic research I discussed previously, many papers in AI capture some new game-theoretical concepts and aim to prove something. In this area are related topics, such as negotiation (and to some extent, argumentation, although that is more related with non-monotonic logic). For some game-theoretical papers that relax assumptions of heterogeneity in the participants involved in the game (for example), simulation is used to find support for properties that cannot feasibly be proven (often, properties with assumptions of homogonous participants are proven and simulation supplements these results with experiments on trickier cases).
In short, I see no problem with you moving to AI, we have plenty of mathematicians (and philosophers and computer scientists) in the field. In terms of logic or game-theory, you may want to pick the topic carefully if you want to avoid spending time learning 'theoretical' Comp. Sci and/or programming.