File:PrimeNumberTheorem.svg

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Original file(SVG file, nominally 500 × 306 pixels, file size: 22 KB)

Summary

Displayed is

  • the number of primes less than or equal to x (red)
  • an approximation using <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24c33f8a1368725ed58e7c3b227d2ab8b638ba9c" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -2.005ex; width:4.524ex; height:4.843ex;" alt="{\displaystyle {\frac {x}{\ln {x}}}}"> (green)
  • an approximation using the logarithmic integral <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4505cf80ca21b5c1464fc13a5f691acec09b487" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:5.291ex; height:2.843ex;" alt="{\displaystyle \operatorname {Li} (x)}"> (blue)

Created using Mathematica:

Plot[{PrimePi[x], x/Log[x], LogIntegral[x] - LogIntegral[2]}, {x, 0, 100000}, PlotStyle -> {Red, Green, Blue}]

(An SVG recreation of File:PrimeNumberTheorem.png from Wikimedia Commons.)

Licensing

File history

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Date/TimeThumbnailDimensionsUserComment
current22:12, 13 January 2017Thumbnail for version as of 22:12, 13 January 2017500 × 306 (22 KB)127.0.0.1 (talk)Displayed is <dl><dd><ul> <li>the number of primes less than or equal to x (red)</li> <li>an approximation using <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>x</mi><mrow><mi>ln</mi><mo>⁡<!-- ⁡ --></mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi></mrow></mrow></mfrac></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\frac {x}{\ln {x}}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24c33f8a1368725ed58e7c3b227d2ab8b638ba9c" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -2.005ex; width:4.524ex; height:4.843ex;" alt="{\displaystyle {\frac {x}{\ln {x}}}}"></span> (green)</li> <li>an approximation using the logarithmic integral <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>Li</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} (x)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4505cf80ca21b5c1464fc13a5f691acec09b487" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:5.291ex; height:2.843ex;" alt="{\displaystyle \operatorname {Li} (x)}"></span> (blue)</li> </ul></dd></dl> <p>Created using Mathematica: </p> <pre>Plot[{PrimePi[x], x/Log[x], LogIntegral[x] - LogIntegral[2]}, {x, 0, 100000}, PlotStyle -> {Red, Green, Blue}] </pre> (An SVG recreation of File:PrimeNumberTheorem.png from Wikimedia Commons.)
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