21 Mar

File:Prime number theorem ratio convergence.svg

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Revision as of 13:08, 21 March 2013
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== {{int:filedesc}} ==

== {{int:filedesc}} ==

{{Information

{{Information
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|Description ={{en|1=A plot showing how two estimates described by the prime number theorem, $\frac{n}{\ln n}$ and $\int_2^n \frac{1}{\ln t} \mathrm{d}t = Li(n) = li(n) - li(2)$ converge asymptotically towards $\pi(n)$ , the number of primes less than n. The x axis is $n$ and is logarithmic (labelled in evenly spaced powers of 10), going up to 10²⁴, the largest $n$ for which $\pi(n)$ is currently known. The former estimate converges extremely slowly, while the latter has visually converged on this plot by 10⁸. Source used to generate this chart is shown below.}}
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|Description ={{en|1=A plot showing how two estimates described by the prime number theorem, $\frac{x}{\ln x}$ and $\int_2^x \frac{1}{\ln t} \mathrm{d}t = Li(x) = li(x) - li(2)$ converge asymptotically towards $\pi(x)$ , the number of primes less than x. The x axis is $x$ and is logarithmic (labelled in evenly spaced powers of 10), going up to 10²⁴, the largest $x$ for which $\pi(x)$ is currently known. The former estimate converges extremely slowly, while the latter has visually converged on this plot by 10⁸. Source used to generate this chart is shown below.}}

|Source ={{own}}

|Source ={{own}}

|Author =[[User:Dcoetzee]]

|Author =[[User:Dcoetzee]]
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LaTeX source for labels:

LaTeX source for labels:



   


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  $$ \left.{\pi(n)}\middle/{\frac{n}{\ln n}}\right. $$

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  $$ \left.{\pi(x)}\middle/{\frac{x}{\ln x}}\right. $$

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  $$ \left.{\pi(n)}\middle/{\int_2^n \frac{1}{\ln t} \mathrm{d}t}\right. $$

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  $$ \left.{\pi(x)}\middle/{\int_2^x \frac{1}{\ln t} \mathrm{d}t}\right. $$

These were converted to SVG with [http://www.tlhiv.org/ltxpreview/] and then the graph was embedded into the resulting document in Inkscape. Axis fonts were also converted to Liberation Serif in Inkscape.

These were converted to SVG with [http://www.tlhiv.org/ltxpreview/] and then the graph was embedded into the resulting document in Inkscape. Axis fonts were also converted to Liberation Serif in Inkscape.

http://commons.wikimedia.org/w/index.php?title=File:Prime_number_theorem_ratio_convergence.svg&diff=93014538&oldid=93014467