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File:Prime number theorem ratio convergence.svg

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Description
English: A plot showing how two estimates described by the prime number theorem, and converge asymptotically towards , the number of primes less than x. The x axis is and is logarithmic (labelled in evenly spaced powers of 10), going up to 1024, the largest for which 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.
Date
Source Own work
Author Dcoetzee
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Source

All source released under CC0 waiver.

Mathematica source to generate graph (which was then saved as SVG from Mathematica):

(* Sample both functions at 600 logarithmically spaced points between \
1 and 2^40 *)
base = N[E^(24 Log[10]/600)];
ratios = Table[{Round[base^x], 
    N[PrimePi[Round[base^x]]/(base^x/(x*Log[base]))]}, {x, 1, 
    Floor[40/Log[2, base]]}];
ratiosli = 
  Table[{Round[base^x], 
    N[PrimePi[
       Round[base^x]]/(LogIntegral[base^x] - LogIntegral[2])]}, {x, 
    Ceiling[Log[base, 2]], Floor[40/Log[2, base]]}];
(* Supplement with larger known PrimePi values that are too large for \
Mathematica to compute *)
LargePiPrime = {{10^13, 346065536839}, {10^14, 3204941750802}, {10^15,
     29844570422669}, {10^16, 279238341033925}, {10^17, 
    2623557157654233}, {10^18, 24739954287740860}, {10^19, 
    234057667276344607}, {10^20, 2220819602560918840}, {10^21, 
    21127269486018731928}, {10^22, 201467286689315906290}, {10^23, 
    1925320391606803968923}, {10^24, 18435599767349200867866}};
ratios2 = 
  Join[ratios, 
   Map[{#[[1]], N[#[[2]]]/(#[[1]]/(Log[#[[1]]]))} &, LargePiPrime]];
ratiosli2 = 
  Join[ratiosli, 
   Map[{#[[1]], N[#[[2]]]/(LogIntegral[#[[1]]] - LogIntegral[2])} &, 
    LargePiPrime]];
(* Plot with log x axis, together with the horizontal line y=1 *)
Show[LogLinearPlot[1, {x, 1, 10^24}, PlotRange -> {0.8, 1.25}], 
 ListLogLinearPlot[{ratios2, ratiosli2}, Joined -> True], 
 LabelStyle -> FontSize -> 14]

LaTeX source for labels:

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

These were converted to SVG with [1] and then the graph was embedded into the resulting document in Inkscape. Axis fonts were also converted to Liberation Serif in Inkscape.

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21 March 2013

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Date/TimeThumbnailDimensionsUserComment
current13:07, 21 March 2013Thumbnail for version as of 13:07, 21 March 2013250 × 160 (87 KB)DcoetzeeChange n to x to match article
12:30, 21 March 2013Thumbnail for version as of 12:30, 21 March 2013250 × 160 (86 KB)DcoetzeeConvert formula from graphics to pure SVG using http://www.tlhiv.org/ltxpreview/
12:23, 21 March 2013Thumbnail for version as of 12:23, 21 March 2013250 × 160 (130 KB)Dcoetzee{{Information |Description ={{en|1=A plot showing how two estimates described by the prime number theorem, <math>\frac{n}{\ln n}</math> and <math>\int_2^n \frac{1}{\ln t} \mathrm{d}t = Li(n) = li(n) - li(2)</math> converge asymptotically towards <ma...

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