Is the distribution of prime numbers chaotic?

Question

The distribution of prime numbers, at first glance, appears to be somewhat random. It is however, deeply structured and deterministic. Does this qualify it as chaotic.

As chaotic is somewhat of a loose term, I will use the following definition: when the initial conditions determine later conditions, but the approximate initial conditions do not approximately determine later conditions. To be more clear, I will break this question down into three parts:

1. Can particular "conditions" be identified for prime numbers? Again this is hard to answer, but maybe there are similar distributions that prime numbers can be compared to.

This aleads to the next question,

2. Do the initial conditions determine the later conditions?

And

3. Do approximate initial conditions not approximate later conditions

Answer 1

As Gerry Myerson already noted in a comment, to talk about the chaoticity of anything, you first have to embed it into the framework of dynamical systems. At the very least, you have to have:

1. some notion of state of a system (this may be what you understand by condition),
2. some dependence of future states on present states,
3. some notion of distance between possible states.

Applying the first point to prime numbers already requires making some very loose associations creativity. There are no different states of prime numbers; they just are.

You could consider an individual prime number as a state of a system and the function $n$ that maps a prime number to the next prime number as your temporal relation between states. But then all you can modify about your sequence is where in the sequence you start – the sequence itself is fixed. The best approximations of a state $p_i$ are its predecessor $p_{i-1}$ and successor $p_{i+1}$. If you insist continuing with this, the best approximations of the present state do allow for the best approximation of the future states. For example, if the present state is $p_i$ the future state after $k$ iterations is $n^k(p_i) = p_{i+k}$ and the respective future states the best approximations of the present state are $p_{i+k-1}$ and $p_{i+k+1}$, which happen to be the best approximations of the future state. But keep in mind that this is already stretching the perspective a lot.

Is the distribution of prime numbers chaotic?

What is that even supposed to mean? I never heard anybody apply the term chaotic to distributions. It’s dynamics that can be chaotic.