Ringkasan

DeskripsiPrime number theorem ratio convergence.svg	English: A plot showing how two estimates described by the prime number theorem, ${\displaystyle {\frac {x}{\ln x}}}$ and ${\displaystyle \int _{2}^{x}{\frac {1}{\ln t}}\mathrm {d} t=Li(x)=li(x)-li(2)}$ converge asymptotically towards ${\displaystyle \pi (x)}$, the number of primes less than x. The x axis is ${\displaystyle x}$ and is logarithmic (labelled in evenly spaced powers of 10), going up to 1024, the largest ${\displaystyle x}$ for which ${\displaystyle \pi (x)}$ 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.
Tanggal	21 Maret 2013
Sumber	Karya sendiri
Pembuat	Dcoetzee
SVG genesis InfoField	The SVG code is valid.   Gambar vektor ini dibuat menggunakan Mathematica    This chart uses embedded text that can be easily translated using a text editor.

Lisensi

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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.