论文标题
分为半段
Partitions into semiprimes
论文作者
论文摘要
令$ \ mathbb {p} $表示素数集，$ \ mathcal {n} \ subset \ subset \ mathbb {n} $是一组，其元素附加了任意权重。设置$ \ mathfrak {p} _ {\ mathcal {n}}（n）$作为限制分区函数，该函数计数$ n $的分区，其所有零件都位于$ \ mathcal {n} $中。通过采用耐铁木圆圈方法的合适变化，我们提供了$ \ mathfrak {p} _ {\ Mathcal {n}}（n）$的渐进公式（计数因素，重复因素计数以及不同的因素）。为了处理次要的弧，我们研究了双重素质的双重脉冲总和，并找到其相应的界限，从而扩展了Vinogradov在分区上的一些结果。我们还描述了一种方法，可以通过为一般加权集$ \ mathcal {n} $找到不同的策略，以为主要的，非主要的主要专业和次要弧线分配不同的策略，从而找到渐近分区$ \ mathfrak {p} _ {\ Mathcal {n}}（n）$。我们的结果与分区渐近学中的其他最新结果一起进行了上下文化。
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.
