Formulas for pi(n) and the n-th prime
Number Theory
2014-10-21 v3 History and Overview
Abstract
Using inequalities of Rosser and Schoenfeld, we prove formulas for pi(n) and the n-th prime that involve only the elementary operations +,-,/ on integers, together with the floor function. Pascal Sebah has pointed out that the formula for pi(n) operates in O(n^(3/2)) time. Similar formulas were proven using Bertrand's Postulate by Stephen Regimbal, An explicit formula for the k-th prime number, Mathematics Magazine, 48 (1975), 230-23
Keywords
Cite
@article{arxiv.math/0210312,
title = {Formulas for pi(n) and the n-th prime},
author = {Sebastian Martin Ruiz and Jonathan Sondow},
journal= {arXiv preprint arXiv:math/0210312},
year = {2014}
}
Comments
4 pages; similar formulas were proven using Bertrand's Postulate by S. Regimbal, An explicit formula for the k-th prime number, Math. Mag., 48 (1975), 230-232