Summing μ ( n ) : a faster elementary algorithm.

IF 0.6 Q3 MATHEMATICS

Research in Number Theory Pub Date : 2023-01-01 DOI:10.1007/s40993-022-00408-8

Harald Andrés Helfgott, Lola Thompson

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引用次数: 0

Abstract

We present a new elementary algorithm that takes $time O_{ϵ} (x^{\frac{3}{5}}, {(log x)}^{\frac{8}{5} + ϵ}) and space O (x^{\frac{3}{10}}, {(log x)}^{\frac{13}{10}})$ (measured bitwise) for computing $M (x) = \sum_{n \leq x} μ (n),$ where $μ (n)$ is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $O (x^{1 / 5} {(log x)}^{5 / 3})$ by the use of (Helfgott in: Math Comput 89:333-350, 2020), at the cost of letting time rise to the order of $x^{3 / 5} {(log x)}^{2} log log x$ .

Abstract Image

查看原文本刊更多论文

求和μ (n):一个更快的初等算法。

我们提出了一种新的初等算法，用于计算M (x) =∑n≤x μ (n)，该算法花费时间O λ x 35 (log x) 85 + λ和空间O λ 3 10 (log x) 13 10(按位测量)，其中μ (n)是Möbius函数。这是1985年以来初等算法对x指数的第一次改进。我们还表明，通过使用(Helfgott in: Math computer 89:333- 350,2020)，可以将空间消耗减少到O (x 1 / 5 (log x) 5 / 3)，代价是让时间上升到x 3 / 5 (log x) 2 log log x的数量级。

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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来源期刊

Research in Number Theory MATHEMATICS-

CiteScore

0.80

自引率

12.50%

发文量

期刊介绍： Research in Number Theory is an international, peer-reviewed Hybrid Journal covering the scope of the mathematical disciplines of Number Theory and Arithmetic Geometry. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to these research areas. It will also publish shorter research communications (Letters) covering nascent research in some of the burgeoning areas of number theory research. This journal publishes the highest quality papers in all of the traditional areas of number theory research, and it actively seeks to publish seminal papers in the most emerging and interdisciplinary areas here as well. Research in Number Theory also publishes comprehensive reviews.