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- W4312206432 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." @default.
- W4312206432 created "2023-01-04" @default.
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- W4312206432 date "2022-12-23" @default.
- W4312206432 modified "2023-09-27" @default.
- W4312206432 title "Partitions into semiprimes" @default.
- W4312206432 doi "https://doi.org/10.48550/arxiv.2212.12489" @default.
- W4312206432 hasPublicationYear "2022" @default.
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