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Non-Abelian Generalizations of the Erdős-Kac Theorem
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- Journal:
- Canadian Journal of Mathematics / Volume 56 / Issue 2 / 01 April 2004
- Published online by Cambridge University Press:
- 20 November 2018, pp. 356-372
- Print publication:
- 01 April 2004
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- Article
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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.
