The VC-Dimension of a class of multiples of the primes and a connection to AdaBoost

We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical risk minimization rule as well as the weights assigned to the output hypothesis in AdaBoost for these “prime-identifying” indicator functions, when we sample \( mn \) i.i.d. points uniformly from the integers \(\{2, \ldots, n\}\).