Topics in Number Theory

PI建议扩展最近与Belabas和Bhargava在有理数域上的给定次复数内的代数数域的数量，以及在给定界下的判别式的绝对值的工作。这项工作也有算法的一面，其中一个计算准确的数字，而不仅仅是一个主要的项与误差估计。此外，PI还提出了对乘阶函数混沌行为的研究。这里我们研究了一个算术函数，它赋予一个奇自然数2的乘阶与这个模数（有明显的推广）。这个函数在很多问题中都是固有的，在密码学中有着重要的应用。在这些项目和提案中的其他项目中，PI将包括研究生、本科生和初级教师，就像他过去所做的那样成功。有理数（分数）是非常基础的，小学就开始教了。在过去的几个世纪里，数论学家已经找到了将它们扩展到更大的领域的自然方法，这些领域涉及到特定多项式的根。对这些新领域进行分类的程序是由高斯在两个多世纪前开始的，在那里他能够解决多项式有二次的情况下的问题。在过去的一个世纪里，我们学会了如何做到这一点，在过去的十年里，我们学会了如何做到这一点。在这项工作的基础上，PI建议从统计的角度和算法的角度对这些领域进行更深入的研究。我们很少有与高等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，可以作为计算工具来开发。此外，PI还建议研究某些在密码学中很重要的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，长度可以有很大的不同）。对于这些周期的正常长度和平均长度，已经提出了一些猜测。PI希望解决其中的一些猜想，也许是否定的。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。提出的问题是根本性的，有些问题已经研究了几个世纪。最近的一些问题与密码学交织在一起，这是一个在世界经济中非常重要的话题。

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.