Topics in Number Theory

PI建议扩展Belabas和Bhargava最近的工作，计算在有理数域上给定度数的复数内代数数域的数量，并且判别式的绝对值低于给定的界限。 这项工作也有一个算法的一面，其中一个计数的确切数字，而不仅仅是一个主项与误差估计。 此外，PI提出研究乘性阶函数的混沌行为。 在这里研究的算术函数分配给一个奇数自然数的乘法阶2这个模数（具有明显的推广）。 这个函数在许多问题中都是固有的，它在密码学中有重要的应用。 在这些项目和提案中的其他项目中，PI将让研究生、本科生和初级教师参与进来，就像他过去成功地做的那样。有理数（分数）是如此基本，以至于在小学里都要教授。 在过去的几个世纪里，数论学家们已经找到了将它们扩展到更大的域的自然方法，这些方法包括引入特定多项式的根。 对这些新领域进行分类的程序是高斯在两个多世纪前开始的，在那里他能够解决多项式具有二阶的情况下的问题。 在过去的世纪里，我们学会了如何为三级学校做这件事，在过去的十年里，我们学会了为四级和五级学校做这件事。 在这项工作的基础上，PI从统计学的角度和算法的角度对这些领域进行了更深入的研究。 我们几乎没有与更高学位领域相关的硬数据，而计算它们的新视角似乎已经成熟，可以作为计算工具开发。 此外，PI建议研究在密码学中重要的某些循环过程的周期长度。 这些长度具有混沌行为（对于附近的参数，长度可以大不相同）。 对于这些周期的正常长度和平均长度，已经提出了一些建议。 PI希望解决其中的一些问题，也许是负面的。 在这些项目和提案中的其他项目中，PI将涉及本科生，研究生和初级教师。 提出的问题是基本的，有些已经研究了几个世纪。 最近的一些问题与密码学有关，密码学是世界经济中非常重要的主题。

The PI proposes to extend recent work with Belabas and Bhargava on counting the number of algebraic number fields within the complex numbers of a given degree over the field of rationals, and with the absolute value of the discriminant below a given bound. This work also has an algorithmic side where one counts the exact number and not just a main term with an error estimate. In addition, the PI proposes to study the chaotic behavior of the multiplicative order function. Here one studies the arithmetic function which assigns to an odd natural number the multiplicative order of 2 to this modulus (with obvious generalizations). This function appears intrinsically in many problems, and it has important applications in cryptography. In these projects and others in the proposal, the PI will involve graduate students, undergraduates, and junior faculty, as he has done with success in the past.The rational numbers (fractions) are so basic that they are taught in elementary school. For the past few centuries, number theorists have found natural ways of expanding them to larger domains that involve throwing in roots of particular polynomials. The program to classify these new fields was begun by Gauss over two centuries ago, where he was able to solve the problem for the case where the polynomial has degree two. In the past century we learned how to do this for degree three, and in the past decade for degrees four and five. Building on this work, the PI proposes a deeper study of these fields from a statistical point of view, and also an algorithmic point of view. We have very few hard data in connection with higher degree fields, and the new perspectives for counting them seem ripe for development as computational tools. In addition, the PI proposes to study the lengths of the periods of certain cyclic processes that are important in cryptography. These lengths have chaotic behavior (for nearby parameters, the lengths can be wildly different). There have been certain conjectures proposed for both the normal lengths of these cycles, and the lengths on average. The PI hopes to settle some of these conjectures, perhaps in the negative. In these projects and others in the proposal, the PI will involve undergraduates, graduate students, and junior faculty. The problems proposed are fundamental and some have been studied for centuries. Some of the more recent problems are entwined with cryptography, a topic of great importance in the world's economy.