Partitions into semiprimes

2022/12/23 by Madhuparna Das, Nicolas Robles, Das, Madhuparna +5 · 2 citations
Mathematics · #11L03 #11L20. Secondary: 11M41 #11P82 #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary: 11P55

paper · pdf · doi:10.48550/arxiv.2212.12489

openalex publication_date 2022/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℙ denote the set of primes and N⊂ ℕ be a set with arbitrary weights attached to its elements. Set \mathfrakp_N(n) to be the restricted partition function which counts partitions of n with all its parts lying in N. By employing a suitable variation of the Hardy-Littlewood circle method we provide the asymptotic formula of \mathfrakp_N(n) for the set of semiprimes N = \p₁ p₂ : p₁, p₂ ∈ 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 \mathfrakp_N(n) for general weighted sets 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.

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