Let $\mathbb{ P}$ denote the set of primes and $\mathcal{ N}\subset \mathbb{ N}$ be a set with arbitrary weights attached to its elements. Set $\mathfrak{ p}_{ \mathcal{ N}}(n)$ to be the restricted partition function which counts partitions of $n$ with all its parts lying in $\mathcal{ N}$. By employing a suitable variation of the Hardy-Littlewood circle method we provide the asymptotic formula of $\mathfrak{ p}_{ \mathcal{ N}}(n)$ for the set of semiprimes $\mathcal{ N} = \{ p_1 p_2 : p_1, p_2 \in \mathbb{ P}\}$ in different set-ups (counting factors, repeating the count of factors, and different factors). In order to deal with the minor arc, we investigate a double Weyl sum over prime products and find its corresponding bound thereby extending some of the results of Vinogradov on partitions. We also describe a methodology to find the asymptotic partition $\mathfrak{ p}_{ \mathcal{ N}}(n)$ for general weighted sets $\mathcal{ N}$ by assigning different strategies for the major, non-principal major, and minor arcs. Our result is contextualized alongside other recent results in partition asymptotics.
Categories: math.NT