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# Non-Abelian Generalizations of the Erd\H os-Kac Theorem

Published:2004-04-01
Printed: Apr 2004
• M. Ram Murty
• Filip Saidak
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## Abstract

Let $a$ be a natural number greater than $1$. Let $f_a(n)$ be the order of $a$ mod $n$. Denote by $\omega(n)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erd\"os and Pomerance: The number of $n\leq x$ coprime to $a$ satisfying $$\alpha \leq \frac{\omega(f_a(n)) - (\log \log n)^2/2 }{ (\log \log n)^{3/2}/\sqrt{3}} \leq \beta$$ is asymptotic to $$\left(\frac{ 1 }{ \sqrt{2\pi}} \int_{\alpha}^{\beta} e^{-t^2/2}dt\right) \frac{x\phi(a) }{ a},$$ as $x$ tends to infinity.
 Keywords: Tur{\' a}n's theorem, Erd{\H o}s-Kac theorem, Chebotarev density theorem, Erd{\H o}s-Pomerance conjecture
 MSC Classifications: 11K36 - Well-distributed sequences and other variations 11K99 - None of the above, but in this section

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