代数整数的性质与算术函数的均值估计及应用

代数整数性质的相关问题和无理数的有理逼近问题一直是数论研究中的重点和难点，有不少相关问题长期悬而未决，这类问题的研究具有重要的理论意义和科学价值。事实上，利用计算结果对研究对象进行理论分析已成为数论研究的重要手段。在本项目中我们将对代数整数的重要测度进行计算和分析，进而对与其相关的重要问题进行讨论，帮助我们更深入地了解代数整数的性质；同时对一些重要无理数的无理测度进行计算，讨论这些无理数的有理逼近问题，为实数的整数化表示理论和算法研究提供理论依据和算法基础。另一方面，算术函数的均值估计及应用一直是解析数论的核心问题，众多著名的数论问题最终都归结为某个算术函数的均值估计，特别是其在整数的因子分布研究中的应用，将帮助我们更好地理解自然数的乘性结构，这也将是本项目研究的重要内容。本项目将计算数论与解析数论进行有机地结合，两者相互促进，共同发展。项目中的理论研究和算法研究都具有重要的科学意义。

The problems on the properties of algebraic integers and on the rational approximations of some irrational numbers have been focused in the Number Theory and attracted much attention. Many relevant problems are hard and have been open for a long time. The importance to study these problems is both in theory and in applications. Using computational results to analyze and study some number theory problems becomes more and more important. One of our main research objectives in this research project is to study the problems on the properties of algebraic integers with computational method. We will also study the irrationality measures of some important irrational numbers, to provide theoretical and arithmetical supports to the rational representations of the real numbers. On the other hand, the estimations of the mean values of arithmetic functions are always important problems in Analytic Number Theory, since many problems in Number Theory can be ascribed to the estimations of the mean values of some arithmetic functions. Their applications on the distribution of divisors of integers will help us to understand better the multiplicative structure of natural numbers. All these problems in this project will be studied with both computational method and analytical method. The research in this project has important scientific significance in both theoretical and computational aspects.