Integers With A Predetermined Prime Factorization

A classic question in analytic number theory is to find asymptotics for $\sigma_{k}(x)$ and $\pi_{k}(x)$, the number of integers $n\leq x$ with exactly $k$ prime factors, where $\pi_{k}(x)$ has the added constraint that all the factors are distinct. This problem was originally resolved by Landau in 1900, and much work was subsequently done where $k$ is allowed to vary. In this paper we look at a similar question about integers with a specific prime factorization. Given $\boldsymbol{\alpha}\in\mathbb{N}^{k}$, $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},...,\alpha_{k})$ let $\sigma_{\boldsymbol{\alpha}}(x)$ denote the number of integers of the form $n=p_{1}^{\alpha_{1}}... p_{k}^{\alpha_{k}}$ where the $p_{i}$ are not necessarily distinct, and let $\pi_{\boldsymbol{\alpha}}(x)$ denote the same counting function with the added condition that the factors are distinct. Our main result is asymptotics for both of these functions.