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Let $\omega(n)$ be the number of distinct prime factors of a natural number $n$.

Note that $\omega(n)=0\iff n=1$, and that $\omega(24)=\omega(2^3\cdot 3^1)=2\ (\not = 4)$.

(For more details, you can see "Distinct Prime Factors" on wolfram math world)

Then, for $m\in\mathbb N$, let $S_m=\{n|n\in\mathbb N, \omega(n)=m\}$. Also, Let $N_m(x)$ be the counting function that gives the number of the elements of $S_m$ less than or equal to $x$, for any real number $x$. See the line graph below. This graph shows $N_m(x)\ (m=1,2,3,4)$ in $1\lt x\le 2309=2\cdot3\cdot5\cdot7\cdot11-1.$ Note that $N_m(2309)=0$ for $m\ge 5$, and that $\omega(2310)=\omega(2\cdot 3\cdot 5\cdot 7\cdot 11)=5$.

$\ \ \ \ \ \ \ \ \ \ $enter image description here

In addition to this, let $P_m(x)=\frac{N_m(x)}{x}\times 100$. See the line graph below. This graph shows $P_m(x)\ (m=1,2,3,4)$ in $1\lt x\le 2309=2\cdot3\cdot5\cdot7\cdot11-1.$

$\ \ \ \ \ \ \ \ \ \ $enter image description here

Then, here are my questions.

Question : Are the followings true?

$(1)$ For every $m\in\mathbb N$, $N_m(x)=O(x)\ (x\to\infty)$.

$(2)$ For every $m\in\mathbb N$, $\lim_{x\to\infty} P_m(x)=0$.

$(3)$ If $x\ge 50$, then $P_2(x)\gt P_m(x)$ for every $m\ge 3$.

The above graphs led me to the these conjectures, but I don't have any good ideas. I would like to know any relevant references. Can anyone help?

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1 Answer 1

up vote 6 down vote accepted

I don't quite understand question (1). Isn't $N_m(x) \leq x$ trivially? You seem to be interested in fixed $m$. Then an affirmative answer to (2) and a negative answer to (3) follows from a classical extension of the prime number theorem due to Landau. See, for example,

http://www2.imperial.ac.uk/~bin06/M34PM16-Analytic-Number-Theory/m3pm16div.pdf

In fact, the answer to (2) is "yes, uniformly in $m$." Indeed, a theorem of Hardy and Ramanujan implies that $N_m(x) \ll x/\sqrt{\log\log{x}}$ uniformly in $m$.

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Thank you very much for this answer! I think what I was trying to ask as $(1)$ is actually $(2)$. –  mathlove Nov 24 '13 at 5:01

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