Implication of Prime Number Theorem to prove a result related to $d_n =\operatorname{lcm}\left\{ 1 ,2,\ldots,n \right\}$

Question

It is known that prime number theorem implies $\lim _{n \to + \infty }\frac {\log(d_n)}{n} =1$.

I am given an exercise on proving $\lim _{n \to + \infty } \frac{\log D_{2n}^{ a+b-1}} {n} = 2(a+b-1)$ assuming prime number theorem but I have no idea how to do it. ($a$, $b$ belongs to integers) .

Can someone please help.

Answer 1

I'm assuming you mean the limit

$$\lim_{n\to \infty} \frac{\log d_{2n}^{a+b-1}}{n}.$$

Now, notice that

$$\frac{\log d_{2n}^{a+b-1}}{n} = 2(a+b-1) \frac{\log d_{2n}}{2n}.$$

Taking $n \to \infty$ yields

$$\lim_{n\to \infty} 2(a+b-1) \frac{\log d_{2n}}{2n} = 2(a+b-1).$$