$\lim_{n\to+\infty} \frac{d_n}{n}$

Question

Question

$\lim_{n\to+\infty} \frac{d_n}{n}$, where $d_n$ is the number of divisors of $n$

Draft I think it doesn't exist

Another somewhat related question, in factoring a number into prime factors, say $n=p_1^{k_1} p_2^{k_2} \cdots p_m^{k_m}$, we can say that $k_i \in \{0,1, 2, \cdots, n\}$?

Answer 1

Hint. Note that $d_n\leq 2\sqrt{n}$: if $a$ is a positive divisor of $n$, then $b=n/a$ is also a positive divisor of $n$ which implies that one of them is $\leq\sqrt{n}$.

So the required limit exists and it is equal to...