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Датотека:Prime number theorem absolute error.svg

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Originalna datoteka ‎(SVG datoteka, nominalno 283 × 178 piksela, veličina: 94 kB)

Opis izmene

Opis
English: A log-log plot showing the absolute error of two estimates to the prime-counting function , given by and . 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 y axis is also logarithmic, going up to the absolute error of at 1024. The error of both functions appears to increase as a power of , with Li(x)'s power being smaller; both clearly diverge. The error of Li(x) appears to smooth out after 109 but this is an artifact due to less data availability for in the larger region. Source used to generate this chart is shown below.
Datum
Izvor Sopstveno delo
Autor Dcoetzee
SVG genesis
InfoField
 
The SVG code is valid.
 
This trigonometry was created with Mathematica.
 
and with Inkscape.
 
 This trigonometry uses embedded text that can be easily translated using a text editor.
Izvorni kod
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Mathematica code

base = N[][10]/600)];
diffs = Table[][base^x], 
    N[][][base^x] - (base^x/(x*Log[base]))]}, {x, 1, 
    Floor[][2, base]}];
diffsli = 
  Table[][base^x], 
    N[][][base^x] - (LogIntegral[base^x] - LogIntegral[2])]}, {x, 
    Ceiling[][base, 2], Floor[][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}};
diffs2 = Abs[][][][[1]], N[][[2]]] - (#[[1]]/(Log[][[1]]]))} &, 
     LargePiPrime]]];
diffsli2 = 
  Abs[][][][[1]], 
       N[][[2]]] - (LogIntegral[][[1]]] - LogIntegral[2])} &, 
     LargePiPrime]]];
(* Plot with log x axis, together with the horizontal line y=1 *)
Show[][1, {x, 1, 10^24}, PlotRange -> {1, 10^21}], 
 ListLogLogPlot[{diffs2, diffsli2}, Joined -> True, 
  PlotRange -> {1, 10^21}], LabelStyle -> FontSize -> 14]

LaTeX source for labels code

$$ {\pi(x)} - {\frac{x}{\ln x}} $$
$$ {\int_2^x \frac{1}{\ln t} \mathrm{d}t} - {\pi(x)} $$

Licenciranje

Ja, nosilac autorskog prava nad ovim delom, objavljujem isto pod sledećom licencom:
Creative Commons CC-Zero Ova datoteka je dostupna pod licencom Creative Commons 1.0 Univerzalna – posvećivanje javnom vlasništvu.
Osoba koja je učestvovalo u radu na ovom dokumentu posvetila je delo javnom vlasništvu, odričući se svih prava na to delo širom sveta, po zakonu o autorskim pravima i povezanim ili srodnim zakonskim pravima koje bi imao/imala, u meri dozvoljenoj zakonom. Možete da umnožavate, menjate, raspodeljujete i prilagođavate delo, čak i u komercijalne svrhe, bez traženja dozvole.

Source

All source released under CC0 waiver.

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



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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licenca
Creative Commons CC0 License Serbian (Cyrillic script) (transliteracija)
izvor datoteke Serbian (transliteracija)
sopstveno delo Serbian (transliteracija)
nastanak
21. mart 2013








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trenutna16:47, 21. mart 2013.Minijatura za verziju na dan 16:47, 21. mart 2013.283 × 178 (94 kB)Dcoetzee== {{int:filedesc}} == {{Information |Description    ={{en|1=A log-log plot showing the absolute error of two estimates to the prime-counting function <math>\pi(x)</math>, given by <math>\frac{x}{\ln x}</math> and <math>\int_2^x \frac{1}{\ln t} \mathrm...










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