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For a positive integer $n$, let $\Omega(n)$ be the number of primes dividing $n$, counted with multiplicities (eg $\Omega(5)=1$, $\Omega(8)=\Omega(12)=3$).

For a real number $x$, let $\{x\}\in[0,1)$ be the fractional part of $x$, so that $x-\{x\}$ is an integer.

If $\alpha\in\mathbb R$ is an irrational number, is the sequence $\big(\{\Omega(n)\alpha\}\big)_{n=1}^\infty$ uniformly distributed in $[0,1)$?

A sequence $(a_n)$ taking values in $[0,1)$ is uniformly distributed if for all $k\in\mathbb{Z}\setminus\{0\}$ we have $$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{\displaystyle2\pi ika_n}=0$$

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  • $\begingroup$ If the Erdös-Kac theorem existed in a more refined version, saying that if you look at the distribution of $\Omega(n)$ over a range of $n$'s, then $\Omega(n)−\log\log n$ is close in total variation distance to a normal random variable, then the answer would be yes. $\endgroup$
    – Anthony Quas
    Commented May 7, 2013 at 15:45
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    $\begingroup$ The answer is yes. Look up the chapter on the Selberg-Delange method in Tenenbaum's book "Intro to Analytic and Probabilistic Number Theory". $\endgroup$
    – Dimitris Koukoulopoulos
    Commented May 8, 2013 at 23:49

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