Normal Order of Certain Arithmetic Functions and New Analogues of the Erd\H{o}s-Kac Theorem
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Authors
Kar, Arpita
Date
Type
thesis
Language
eng
Keyword
Number Theory , Probabilistic Number Theory
Alternative Title
Abstract
In 1917, Hardy and Ramanujan planted the seeds of Probabilistic Number Theory when they defined `normal order' of an arithmetic function. This was nurtured further by Erd\H{o}s and Kac and many other great minds who infused further improvements to the probabilistic theory of additive functions. A systematic generalisation of this study to non-additive functions was initiated by Ram Murty and Kumar Murty in 1984. In this thesis, we study the theorem of Hardy and Ramanujan more deeply from a different perspective where we view the normal order results as estimates on sizes of certain exceptional sets. Using these results, we study the normal order of various non-additive functions like $\Omega(\phi(p+a))$ and $\Omega(\tau(p+a))$, thus generalising the results of Murty and Murty to shifts of prime arguments. We also study the distribution of these functions. Here $\Omega(n)$ counts the number of prime factors of a natural number $n$, with multiplicity, $\phi(n)$ denotes the Euler totient function and $\tau(n)$ denotes the Ramanujan tau function. Here and throughout the thesis, we denote $p$ as a prime. Finally, we study the moments of Dirichlet characters and establish normal order results in that direction as well.
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CC0 1.0 Universal
ProQuest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.
CC0 1.0 Universal