<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let π<italic>(x)</italic> be the number of primes not exceeding <italic>x.</italic> We prove that <math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>π</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>&lt;</mo><mfrac><mi>x</mi><mrow><mo>log</mo><mi>x</mi><mo>-</mo><mn>1.006789</mn></mrow></mfrac></mrow></math> for <math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>≥</mo><msup><mi>e</mi><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow></msup></mrow></math>, and that for sufficiently large <math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>:</mo><mtext> </mtext><mfrac><mi>x</mi><mrow><mo>log</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>log</mo><mtext> </mtext><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>-</mo><mn>1.5</mn></mrow></msup><mo>+</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mo>log</mo><mtext> </mtext><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>-</mo><mn>0.5</mn></mrow></msup></mrow></mfrac><mo>&lt;</mo><mi>π</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>&lt;</mo><mfrac><mn>1</mn><mrow><mo>log</mo><mtext> </mtext><mi>x</mi><mo>-</mo><mn>1</mn><mo>-</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mo>log</mo><mtext> </mtext><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>-</mo><mn>0.5</mn></mrow></msup><mo>-</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>log</mo><mtext> </mtext><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>-</mo><mn>1.5</mn></mrow></msup></mrow></mfrac><mo>.</mo></mrow></math> We finally prove that for <math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>≥</mo><msup><mi>e</mi><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>12</mn></mrow></msup></mrow></msup></mrow></math> and <italic>k</italic> = 2, 3,…, 147297098200000, the closed interval [(<italic>k</italic> – 1)<italic>x, kx</italic>] contains at least one prime number, i.e. the Bertrand's postulate holds for <italic>x</italic> and <italic>k</italic> as above.</p></abstract>

Let π(x) be the number of primes not exceeding x. We prove that π(x)<xlogx-1.006789 for x≥e1012, and that for sufficiently large x: xlogx-1+(log x)-1.5+2(log x)-0.5<π(x)<1log x-1-2(log x)-0.5-(log x)-1.5. We finally prove that for x≥e1012 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.

On the Function π(<italic>x</italic>)

Institute of Geodynamics Sabba S. Ştefănescu of Romanian Academy 19-21 Jean-Louis Calderon St., Bucharest-37, Romania

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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Let π(x) be the number of primes not exceeding x. We prove that π(x)<xlogx-1.006789 for x≥e1012, and that for sufficiently large...

On the Function π(x)

Published Online: Dec 10, 2014

Page range: 25 - 33

Received: May 01, 2013

Accepted: Sep 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0002

Keywords
Arithmetic functions, Inequalities, Bertrand's postulate

© 2014 Magdalena Bănescu

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the Function π(x)

Published Online: Dec 10, 2014

Page range: 25 - 33

Received: May 01, 2013

Accepted: Sep 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0002

KeywordsArithmetic functions, Inequalities, Bertrand's postulate

© 2014 Magdalena Bănescu

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Arithmetic functions, Inequalities, Bertrand's postulate