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

这项研究涉及数论中的加法和乘法函数。我们将通过扩充傅立叶分析、筛法和指数和等经典方法来研究它们。我们将从泛函分析中汲取新的观点，将其应用到他们的研究中。泛函分析将以统一的方式在无限的有限维空间集合中进行。除了允许对某些众所周知的结果进行重新解释，例如图兰-库比利乌斯不等式，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，例如一个整数的除数的个数或小于该整数的相对质数的自然数的个数，都是所谓的乘法函数。这项研究将运用分析和概率论的各种工具来研究这类函数的一般性质以及与之密切相关的称为加性函数的函数。

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.