Ամփոփում

ՆկարագրումPrime 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.
Թվական	21 Մարտ 2013
Աղբյուր	Բեռնողի սեփական աշխատանք
Հեղինակ	Dcoetzee
SVG genesis InfoField	Այս մասշտաբավորելի վեկտորային գրաֆիկայի նիշքի ելակետային կոդը վավեր է.   Սա վեկտոր գրաֆիկական պատկեր է ստեղծված է եղել Mathematica օգնությամբ: .    This chart uses embedded text that can be easily translated using a text editor.

Արտոնագրում

Ես, սույն աշխատանքի հեղինակային իրավունքների տերը, այսուհետ այն հրատարակում եմ հետևյալ արտոնագրի ներքո։

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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.