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Non-Abelian Generalizations of the Erdős-Kac Theorem
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $a$ be a natural number greater than 1. Let ${{f}_{a}}\left( n \right)$ be the order of $a\,\bmod \,n$. Denote by $\omega \left( n \right)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance:
The number of $n\,\le \,x$ coprime to a satisfying
is asymptotic to $\left( \frac{1}{\sqrt{2\pi }}\int_{\alpha }^{\beta }{{{e}^{-{{t}^{2}}/2}}}dt \right)\frac{x\phi \left( a \right)}{a}$ as $x$ tends to infinity.
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- Copyright © Canadian Mathematical Society 2004
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