Honors Theses

Date of Award

Spring 5-1-2021

Document Type

Undergraduate Thesis

Department

Mathematics

First Advisor

Rizwanur Khan

Second Advisor

Micah Milinovich

Third Advisor

Ayla Gafni, Lê Thái Hoàng

Relational Format

Dissertation/Thesis

Abstract

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. A celebrated result of Erd{\H o}s and Kac states that $\omega(n)$ as a Gaussian distribution. In this thesis, we establish a weighted version of Erd{\H o}s-Kac Theorem. Specifically, we show that the Gaussian limiting distribution is preserved, but shifted, when $\omega(n)$ is weighted by the $k-$fold divisor function $\tau_k(n)$. We establish this result by computing all positive integral moments of $\omega(n)$ weighted by $\tau_k(n)$.

We also provide a proof of the classical identity of $\zeta(2n)$ for $n \in \mathbb{N}$ using Dirichlet's kernel.

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Included in

Number Theory Commons

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