#### Date of Award

1-1-2022

#### Document Type

Thesis

#### Degree Name

M.S. in Mathematics

#### Department

Mathematics

#### First Advisor

Rizwanur R. Khan

#### Second Advisor

Micah B. Milinovich

#### Third Advisor

Rizwanur R. Khan

#### Relational Format

dissertation/thesis

#### Abstract

The distribution of the prime numbers has intrigued number theorists for centuries. As our understanding of this distribution has evolved, so too have our methods of analyzing the related arithmetic functions. If we let ω(n) denote the number of distinct prime divisors of a natural number n, then the celebrated Erdős –Kac Theorem states that the values of ω(n) are normally distributed (satisfying a central limit theorem as n varies). This result is considered the beginning of Probabilistic Number Theory. We present a modern proof of the Erdős–Kac Theorem using a moment based argument due to Granville and Soundararajan, which we explain in full detail. We also use similar techniques to study the second moment of ω(n), refining a classical result of Turán.

#### Recommended Citation

Derrick, Jacob Mitchell, "ON THE DISTRIBUTION OF THE NUMBER OF PRIME FACTORS OF AN INTEGER" (2022). *Electronic Theses and Dissertations*. 2209.

https://egrove.olemiss.edu/etd/2209