Mathematical Sciences: Arithmetic Functions, Their Value Distribution and Application to Algebra

数学科学：算术函数、其值分布及其在代数中的应用

• 批准号: 8722913
• 负责人: Peter Elliott
• 金额: $ 6.72万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-08-01 至 1992-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8722913&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Arithmetic; Functions; Their

This research concerns additive and multiplicative functions in number theory. They will be studied by augmenting the classical methods of Fourier analysis, sieve methods and exponential sums. New ideas will be taken from functional analysis to apply to their study. The functional analysis is to be carried out in an infinite collection of finite dimensional spaces, in a uniform manner. Besides allowing a re- interpretation of certain well-known results, such as the Turan- Kubilius inequality, this approach offers methods and directions for future development. Most of the classical functions in number theory, such as the number of divisors of an integer or the number of natural numbers less than an integer which are relatively prime to that integer, are what are called multiplicative functions. This research will apply various tools from analysis and probability theory to study the general properties of such functions and the closely related functions called additive functions.

这项研究涉及数论中的加法和乘法函数。 将通过增强傅里叶分析、筛分法和指数和的经典方法来研究它们。 泛函分析中的新想法将应用于他们的研究。 泛函分析将以统一的方式在有限维空间的无限集合中进行。 除了允许重新解释某些众所周知的结果（例如图兰-库比利乌斯不等式）之外，这种方法还为未来的发展提供了方法和方向。 数论中的大多数经典函数，例如整数的约数个数或小于整数且与该整数互质的自然数的个数，都是所谓的乘法函数。 本研究将应用分析和概率论的各种工具来研究此类函数的一般性质以及称为加法函数的密切相关函数。