On the Distribution of Integers with Restricted Prime Factors I

Let $E_0,\ldots,E_n$ be a partition of the set of prime numbers, and define $E_j(x) := \sum_{p \in E_j \atop p \leq x} \frac{1}{p}$. Define $\pi(x;\mathbf{E},\mathbf{k})$ to be the number of integers $n \leq x$ with $k_j$ prime factors in $E_j$ for each $j$. Basic probabilistic heuristics suggest that $x^{-1}\pi(x;\mathbf{E},\mathbf{k})$, modelled as the distribution function of a random variable, should satisfy a joint Poisson law with parameter vector $(E_0(x),\ldots,E_n(x))$, as $x \rightarrow \infty$. We prove an asymptotic formula for $\pi(x;\mathbf{E},\mathbf{k})$ which contradicts these heuristics in the case that for each $j$, $E_j(x)^2 \leq k_j \leq \log^{\frac{2}{3}-\epsilon} x$ for each $j$ under mild hypotheses. As a particular application, we prove an asymptotic formula regarding integers with prime factors from specific arithmetic progressions, which generalizes a result due to Delange.