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See detailThe $n$-th prime asymptotically
Arias de Reyna, Juan; Toulisse, Jérémy UL

in Journal de Théorie des Nombres de Bordeaux (2013)

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the compu- tation of its terms is also given, which will be an improvement of that by Salvy (1994 ... [more ▼]

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the compu- tation 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 re- tained the first $m$ terms, for $1 ≤ m ≤ 11$, are given. Finally, as- suming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n ≥ r_3$ , we have $p_n > s_3 (n)$ where $s_3 (n)$ is the sum of the first four terms of the asymptotic expansion. [less ▲]

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