The n-th prime asymptotically

TL;DR

This paper presents a new derivation of the asymptotic expansion of the n-th prime, introduces a faster algorithm for computing its terms, and provides bounds and estimations related to prime distribution assuming the Riemann Hypothesis.

Contribution

It offers a novel derivation of the prime asymptotic expansion, an improved algorithm for term computation, and bounds on the error and prime estimates under the Riemann Hypothesis.

Findings

New derivation of prime asymptotics

Faster algorithm for computing expansion terms

Bounds on error and prime estimates assuming Riemann Hypothesis

Abstract

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with $\li^{-1}(n)$, after having retained the first m terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n> s_3(n)$ where $s_3(n)$ is the sum of the first four terms of the asymptotic expansion.