Title: New estimates for the $n$th prime number,
Abstract: In this paper we establish a new explicit upper and lower bound for the
$n$-th prime number, which improve the currently best estimates given by Dusart
in 2010. As the main tool we use some recently obtained explicit estimates for
the prime counting function. A further main tool is the usage of estimates
concerning the reciprocal of $\log p_n$. As an application we derive refined
estimates for $\vartheta(p_n)$ in terms of $n$, where $\vartheta(x)$ is
Chebyshev's $\vartheta$-function.