New estimates for the $n$th prime number

TL;DR

This paper provides improved explicit bounds for the $n$-th prime number using recent estimates of the prime counting function and reciprocal logarithm bounds, with applications to Chebyshev's $ heta$-function.

Contribution

It introduces new explicit upper and lower bounds for the $n$-th prime, surpassing previous estimates by Dusart (2010).

Findings

Refined bounds for the $n$-th prime number.

Enhanced estimates for Chebyshev's $ heta$-function.

Improved tools for prime number analysis.

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.