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 Tur{\' a}n's theorem, Erd{\H o}s-Kac theorem, Chebotarev density theorem, Erd{\H o}s-Pomerance conjecture

MSC Classifications:

11K36, 11K99 show english descriptions Well-distributed sequences and other variations
None of the above, but in this section 11K36 - Well-distributed sequences and other variations
11K99 - None of the above, but in this section

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