The n-th prime asymptotically

2012/03/31 by Juan Arias de Reyna, Toulisse Jeremy · 1 citation
Mathematics · #math.NT

paper · pdf · doi:10.5802/jtnb.847

published as J. Théor. Nombres Bordeaux 25 (2013) 521-555

arxiv created 2012/04/05 · arxiv updated 2014/03/25

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≤ m≤ 11, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible r₃ such that, for n≥ r₃, we have p_n> s₃(n) where s₃(n) is the sum of the first four terms of the asymptotic expansion.

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