Topics in Number Theory

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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.