For any integer $n\ge 1$, which of the following is/are true? For any integer $n\ge 1$, let $d(n) =$ number of positive divisors of $n$
$v(n) =$ number of distinct prime divisors of $n$
$\omega(n)$= number of prime divisors of $n$ counted with multiplicities. for example $\omega(p^2)=2$, for a prime $p$.
Then which of the following is/are true
$1.~$ If $n\ge 1000$ and $\omega(n)\ge 2$, then $d(n)> \log~n$
$2.~$ There exist $n$ such that $d(n)>3\sqrt n$
$3.~$ For every $n, ~2^{v(n)}\le d(n) \le 2^{\omega(n)}$
$4.~$ If $\omega(n)=\omega(m),$ then  $d(n)=d(m)$
My try: If $n=p_1^{n_1}\cdots p_k^{n_k}$ be the factorization of $n$, then $d(n)=(n_1+1)\times \cdots \times (n_k+1)$, $v(n)=k $ and $\omega(n)=n_1+\cdots n_k$,
Using this, (4) is false, Counter example, $\omega(2^63^2)=\omega(5^47^4)=8$ but  $d(2^63^2)=21 \ne d(5^47^4)=25$.
I am not able to conclude other options please help.
 A: For option $\bf(1)$, 
Let $~n=37^2>1000~$
Clearly, here $~w(n)=\text{number of prime divisors of $~n~$ counted with multiplicities}=2~$
and $~d(n)=\text{number of positive divisors of $~n~$}=3~$ and also $~\log(n)=\log(37^2)=2\log(37)>3=d(n)~.$
Therefore option $(1)$ is incorrect.
For option $\bf(2)$, 
Let $~n=p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}~,$ then $~d(n)=(r_1+1)(r_2+1)\cdots (r_k+1)~$
Again $~3\sqrt n=3~p_1^{r_1/2}p_2^{r_2/2}\cdots p_k^{r_k/2}\ge d(n)~,~~~\forall ~n\in\mathbb N$
Hence $~\nexists~n\in\mathbb N~,~$ such that $~d(n)>3\sqrt n ~$ and therefore option $(2)$ is again incorrect.
For option $\bf(3)$, 
Let $~n=p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}~,$ then
$~d(n)=(r_1+1)(r_2+1)\cdots (r_k+1)~$
$~w(n)=r_1+r_2+\cdots r_k~$
$~v(n)=\text{number of distinct prime divisors of $~n~$}=k~$
So $~2^{w(n)}=2^{r_1+r_2+\cdots r_k}~$ and $~2^{v(n)=2^k}~$
Clearly, $~2^k\le (r_1+1)(r_2+1)\cdots (r_k+1)\le 2^{r_1+r_2+\cdots r_k}~$
which implies, $~n, ~2^{v(n)}\le d(n) \le 2^{w(n)}~$ and  therefore option $(3)$ is correct.
For option $\bf(4)$, 
Let $~n=9=3^2~$ and $~m=3\times 7=21~,$ then $~w(n)=w(m)=2~$ and $~d(n)=3~, ~~d(m)=4$ 
So although for some $~n~$ and $~m~$, $~w(n)=w(m)~,$ but $~d(n)\ne d(m)~$ and therefore option $(4)$ is also incorrect.
