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Formulas for pi(n) and the n-th prime

Sebastian Martin Ruiz·Jonathan Sondow
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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

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Sebastian Martin Ruiz, Jonathan Sondow. 2014-03-23. Formulas for pi(n) and the n-th prime. https://arxiv.org/abs/math/0210312

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