Number Theory Commons^™

Rough Numbers And Variations On The Erdős--Kac Theorem, Kai Fan

Dartmouth College Ph.D Dissertations

The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory.

In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y>=1. Denoting by Phi(x,y) the count of these numbers up to any given x>=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)<0.6x/\log y holds uniformly for all 3<=y<=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges.

In the second part, we turn …

Counting Elliptic Curves With A Cyclic M-Isogeny Over Q, Grant S. Molnar

Dartmouth College Ph.D Dissertations

Using methods from analytic number theory, for m > 5 and for m = 4, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic m-isogeny up to quadratic twist over the rational numbers. For m > 5, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic m-isogeny up to isomorphism over the rational numbers.

Triangular Modular Curves Of Low Genus And Geometric Quadratic Chabauty, Juanita Duque Rosero

Dartmouth College Ph.D Dissertations

This manuscript consists of two parts. In the first part, we study generalizations of modular curves: triangular modular curves. These curves have played an important role in recent developments in number theory, particularly concerning hypergeometric abelian varieties and approaches to solving generalized Fermat equations. We provide a new result that shows that there are only finitely many Borel-type triangular modular curves of any fixed genus, and we present an algorithm to list all such curves of a given genus.

In the second part of the manuscript, we explore the problem of computing the set of rational points on a smooth, …