In probability/statistics, there is a notion of two things being "independent", which would basically mean that any information we can get about one thing has no effect on our (probabilistic) knowledge of the other.
What are the possible notions of "independent" for the natural numbers? Under such a notion, for instance, the properties of being "multiple of 2" and "multiple of 3" are independent, while "multiple of 4" and "multiple of 6" are not, because something being a multiple of 4 means that it is even and that makes it more likely that it is a multiple of 6.
There's no probability measure on the natural numbers where every natural number carries an equal positive weight (no uniform distribution on the natural numbers, because they're countable) so the substitute seems to be to look at all the natural numbers up to some natural number n, consider the extent of dependence there, and then take the limit as $n \to \infty$. Or, perhaps instead of looking at initial segments, we can look at segments of consecutive integers starting and ending at finite points, and measure the degree of dependence there. Are there other notions that are qualitatively different, or stronger, or weaker? What notions of independence are most useful for specific applications (such as the distribution of prime numbers, additive combinatorics)?
On a related note, is there some way of making sense of the "correlation" between two (infinite) subsets of the natural numbers, that would play some role analogous to what correlation plays in probabliity/statistics? Even if there isn't a numerically rigorous way, is there some way we can define a notion of "uncorrelated" for infinite subsets of the natural numbers. (Hopefully, in a way that independent subsets are uncorrelated)? My guess would be to measure correlations in some suitable way for all numbers up to $n$ and then take the limit as $n \to \infty$.