Do prime numbers satisfy this? Is this true that $n\log\left(\frac{p_n}{p_{n+1}}\right)$ is bounded, where $p_n$ is the $n$-th prime number?

$\frac{p_n}{p_{n+1}}<1$ for all $n$, so when $n\to\infty$, $(\frac{p_n}{p_{n+1}})^n\to0$.
so
$$n\log(\frac{p_n}{p_{n+1}})=\log(\frac{p_n}{p_{n+1}})^n\to-\infty$$
so this is not bounded
