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A Weighted Version Of Erdős-Kac Theorem, Unique Subedi
A Weighted Version Of Erdős-Kac Theorem, Unique Subedi
Honors Theses
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.