Show that $\lim_{n\to\infty}\mathcal I\left(\exp\left(\frac{2\pi\cdot i}{\log_n(p_n\#)}\right)\right)=0$

Question

I'm pretty sure the imaginary part $\mathcal I$ of $f(n)=\exp\left(\frac{2\pi\cdot i}{\log_n(p_n\#)}\right)$ converges as $n\to\infty$, and probably to $0$.

I stumbled upon this result studying the Collatz conjecture and it seemed potentially interesting in respect of prime number theory. Is this a difficult result to prove and does it have any significance?

$f:\Bbb N\to \Bbb C$

$p_n\#$ is the $n^{th}$ primorial

Show that $\lim_{n\to\infty} \mathcal I(f(n))=0$

Answer 1

First, we can rewrite $f(n)$ as

$$f(n)=\sin\left(\frac{2\pi \log n}{\log p_n \#}\right).$$

We can use the fact that $p_n\# =e^{(1+o(1))n\log n}$ to get

$$f(n)=\sin\left(\frac{2\pi}{(1+o(1))n}\right),$$

which, of course, goes to zero as $n$ goes to infinity.