Normal Order of Certain Arithmetic Functions and New Analogues of the Erd\H{o}s-Kac Theorem

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.