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

本研究涉及数论中的加性函数和乘法函数。他们将通过增加傅里叶分析，筛法和指数和的经典方法来研究。新的想法将从功能分析应用到他们的研究。泛函分析是在有限维空间的无限集合中以统一的方式进行的。除了允许对某些众所周知的结果（如图兰-库比留斯不等式）进行重新解释外，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，如整数的除数或小于整数且相对于该整数素数的自然数的数量，都被称为乘法函数。本研究将运用分析和概率论的各种工具来研究这些函数以及与之密切相关的称为加性函数的函数的一般性质。

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.