Partitions into semiprimes

TL;DR

This paper derives an asymptotic formula for the number of partitions of an integer into semiprimes using advanced analytic number theory techniques, extending classical results and providing a methodology for general weighted sets.

Contribution

It introduces a novel application of the Hardy-Littlewood circle method to semiprime partitions and develops bounds for double Weyl sums over prime products, extending Vinogradov's results.

Findings

Asymptotic formula for semiprime partitions derived

Extended bounds for Weyl sums over prime products

Methodology for asymptotic analysis of weighted sets

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.