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Non-Abelian Generalizations of the Erdős-Kac Theorem

Published online by Cambridge University Press:  20 November 2018

M. Ram Murty  and
Filip Saidak
M. Ram Murty
Affiliation:
Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6 e-mail: murty@mast.queensu.ca e-mail: filips@math.ucalgary.ca
Filip Saidak
Affiliation:
Department of Mathematics, Queen's University, Kingston, Ontario, K7L 3N6 e-mail: murty@mast.queensu.ca e-mail: filips@math.ucalgary.ca
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Abstract

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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

$$\alpha \le \frac{\omega \left( {{f}_{a}}\left( n \right) \right)-{{\left( \log \,\log \,n \right)}^{2}}/2}{{{\left( \log \,\log \,n \right)}^{3/2}}/\sqrt{3}}\le \beta $$

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.

Keywords

11K3611K99Turán's theoremErdős-Kac theoremChebotarev density theoremErdős-Pomerance conjecture
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 2 , 01 April 2004 , pp. 356 - 372
Copyright
Copyright © Canadian Mathematical Society 2004

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