Partitions into semiprimes

Abstract

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.

37 pages, 4 figures. Minor corrections and revisions from original version

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