Bounds on the Möbius-signed partition numbers

The Ramanujan Journal Pub Date : 2024-07-08 DOI:10.1007/s11139-024-00885-8

Taylor Daniels

引用次数: 0

Abstract

For $n \in \mathbb {N}$ let $\Pi [n]$ denote the set of partitions of n, i.e., the set of positive integer tuples $(x_1,x_2,\ldots ,x_k)$ such that $x_1 \ge x_2 \ge \ldots \ge x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing $f:\mathbb {N}\rightarrow \{0,\pm 1\}$, for $\pi = (x_1,x_2,\ldots ,x_k) \in \Pi [n]$ let $f(\pi ) := f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the signed partition numbers

$$\begin{aligned} p(n,f) = \sum _{\pi \in \Pi [n]} f(\pi ). \end{aligned}$$

Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for $p(n,\mu )$ and $p(n,\lambda )$, where $\mu $ and $\lambda $ denote the Möbius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities p(n, f) generalize the classical notion of restricted partitions.

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莫比乌斯符号分割数的界限

For （n \in \mathbb {N}\) let $\Pi [n]$ denote the set of partitions of n, i.e..、((x_1,x_2,\ldots,x_k)\)使得（x_1 \ge x_2 \ge \ldots \ge x_k）并且（x_1 + x_2 + \cdots + x_k = n\ ）的正整数元组的集合。固定（f:mathbb {N}\rightarrow \{0,\pm 1\} ），对于（pi = (x_1,x_2,\ldots ,x_k) \in \Pi [n]\) 让（f(\pi ) := f(x_1)f(x_2)\cdots f(x_k)）。这样我们就定义了有符号的分割数 $$\begin{aligned} p(n,f) = \sum _{\pi \in \Pi [n]} f(\pi ).\end{aligned}$$Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for $p(n,\mu )$ and $p(n,\lambda )$, where $\mu $ and$\lambda $ denied the Möbius and Liouville functions from prime number theory, respectively.此外，我们还讨论了量 p(n, f) 如何概括受限分区的经典概念。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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The Ramanujan Journal

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