 Probabilistic number theory

Probabilistic number theory treats an integer chosen uniformly from $1,\dots ,N$ as a random object and studies the distribution of its arithmetic functions. The distribution changes with $N$, and limit theorems describe the behavior as $N$ grows.

For a fixed prime $p$, the probability that $p$ divides the chosen integer is $\lfloor N/p\rfloor /N$, approaching $1/p$. For finitely many distinct primes, the joint divisibility frequencies similarly approach products of these probabilities. This motivates models of the number of prime factors and helps explain why logarithmic scales and normal distributions appear in integer statistics. Such independence is an asymptotic model with specified ranges, not an assumption valid for every growing collection of primes.