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Open Access Dissertation
Doctor of Philosophy (PhD)
Year Degree Awarded
Month Degree Awarded
This doctoral dissertation concerns two problems in number theory. First, we examine a family of discrete dynamical systems in F_2[t] analogous to the 3x + 1 system on the positive integers. We prove a statistical result about the large-scale dynamics of these systems that is stronger than the analogous theorem in Z. We also investigate mx + 1 systems in rings of functions over a family of algebraic curves over F_2 and prove a similar result there.
Second, we describe some interesting properties of zeta functions of algebraic curves. Generally L-functions vanish only to the order required by their root number. However, we demonstrate that for a certain class of quaternion extensions of F_p(t), the zeta function vanishes at a higher order than the root number demands, indicating some other phenomenon at work.
Nichols, Daniel, "Dynamical Systems and Zeta Functions of Function Fields" (2017). Doctoral Dissertations. 941.