Electronic Theses and Dissertations

Date of Award


Document Type


Degree Name

M.S. in Mathematics



First Advisor

Rizwanur R. Khan

Second Advisor

Micah B. Milinovich

Third Advisor

Rizwanur R. Khan

Relational Format



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.



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