Related to bounds for n-th prime

Question

Here we have the bounds:

$$\log n+ \log \log n - 1 < \dfrac {p_n}{n} < \log n+ \log \log n $$, for $n \geq 6$.

Can this constant $1$ be improved in the following way:

For every $1>c>0$ there exists $n(c)$ so that we have

$$\log n+ \log \log n - c < \dfrac {p_n}{n} < \log n+ \log \log n $$, for $n \geq n(c)$.

Of course $c \in \mathbb R$ and $n(c) \in \mathbb N$

Answer 1

The same article as you link says $$\frac{p_n}{n}=\log n+\log\log n-1+o(1).$$ Therefore, for any $c>0$ and large enough $n$, $$\log n+\log\log n-1<\frac{p_n}{n}<\log n+\log\log n-1+c.$$