On the Function π(x)

Bănescu, Magdalena

doi:10.2478/auom-2014-0002

Abstract

Let π(x) be the number of primes not exceeding x. We prove that $π (x) < \frac{x}{log x - 1.006789}$ for $x \geq e^{10^{12}}$ , and that for sufficiently large $x : \frac{x}{log x - 1 + {(log x)}^{- 1.5} + 2 {(log x)}^{- 0.5}} < π (x) < \frac{1}{log x - 1 - 2 {(log x)}^{- 0.5} - {(log x)}^{- 1.5}} .$ We finally prove that for $x \geq e^{10^{12}}$ and k = 2, 3,…, 147297098200000, the closed interval [(k – 1)x, kx] contains at least one prime number, i.e. the Bertrand's postulate holds for x and k as above.

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On the Function π(x)

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