#### Author ORCID Identifier

https://orcid.org/0000-0002-2825-0736

#### Date of Award

Fall 10-1-2023

#### Document Type

Thesis (Ph.D.)

#### Department or Program

Mathematics

#### First Advisor

Carl Pomerance

#### Abstract

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 to the topic of weighted Erdős--Kac theorems for general additive functions. Our results concern the distribution of additive functions f(n) weighted by nonnegative multiplicative functions alpha(n) in a wide class. Building on the moment method of Granville, Soundararajan, Khan, Milinovich and Subedi, we establish uniform asymptotic formulas for the moments of f(n) with a suitable growth rate. Our method also enables us to prove a qualitative result on the moments which extends a theorem of Delange and Halberstam on the moments of additive functions. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem with simple and interesting applications to the Ramanujan tau function and Euler's totient function, the latter of which generalizes an old result of Erdős and Pomerance which shows that as an arithmetic function, the total number of prime factors of values of Euler's totient function satisfies a Gaussian law.

#### Recommended Citation

Fan, Kai, "Rough Numbers and Variations on the Erdős--Kac Theorem" (2023). *Dartmouth College Ph.D Dissertations*. 156.

https://digitalcommons.dartmouth.edu/dissertations/156