On the Distribution of Integers with Restricted Prime Factors I

Let $E_0,\dots ,E_n$ be a partition of the set of prime numbers, and define $E_j(x):={\sum }_{\genfrac{}{}{0px}{}{p\in E_j}{p\le x}}\frac{1}{p}$. Define $\pi (x;\mathbf{E},\mathbf{k})$ to be the number of integers $n\le x$ 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 $(E_0(x),\dots ,E_n(x))$, as $x\to \mathrm{\infty }$. We prove an asymptotic formula for $\pi (x;\mathbf{E},\mathbf{k})$ which contradicts these heuristics in the case that for each $j$, $E_j(x)^2 \leq k_j \leq \log^{\frac{2}{3}-\epsilon} x$foreach$j$ 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.