On the Function π(x)

Magdalena Bănescu

doi:10.2478/auom-2014-0002

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On the Function π(x)

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 22 (2014): Issue 1 (March 2014)

By: Magdalena Bănescu

Open Access

|Dec 2014

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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Articles in this issue

DOI: https://doi.org/10.2478/auom-2014-0002 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 25 - 33

Submitted on: May 1, 2013

Accepted on: Sep 1, 2013

Published on: Dec 10, 2014

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Arithmetic functions,

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© 2014 Magdalena Bănescu, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 22 (2014): Issue 1 (March 2014)