A Weighted Version of Erdős-Kac Theorem

Author

Unique SubediFollow

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.

Recommended Citation

Subedi, Unique, "A Weighted Version of Erdős-Kac Theorem" (2021). Honors Theses. 1679.
https://egrove.olemiss.edu/hon_thesis/1679

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