Slika:Prime number theorem absolute error.svg
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Povzetek
| 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 | |
| Vir | lastno delo |
| Avtor | Dcoetzee |
| SVG razvoj | File:.svg in z Inkscape. |
| Izvorna koda | Mathematica codebase = 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$$ {\pi(x)} - {\frac{x}{\ln x}} $$
$$ {\int_2^x \frac{1}{\ln t} \mathrm{d}t} - {\pi(x)} $$
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Licenca
Jaz, imetnik avtorskih pravic na tem delu, ga objavljam pod naslednjo licenco:
 
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Datoteka je na voljo pod licenco Creative Commons Univerzalna izročitev v javno domeno CC0 1.0 |
| Oseba, ki je delo povezala s tem dovoljenjem, je dala svoje delo v javno domeno z opustitvijo vseh svojih pravic do dela po vsem svetu pod avtorskim pravom, vključno z vsemi povezanimi in sorodnimi pravicami, v obsegu, kot ga dopušča zakonodaja. Delo lahko kopirate, spreminjate, razširjate in izvajate, tudi v gospodarske namene, ne da bi morali zaprositi za dovoljenje.
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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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| Datum in čas | Sličica | Velikost | Uporabnik | Komentar | |
|---|---|---|---|---|---|
| trenutno | 15:47, 21. marec 2013 | | 283 × 178 (94 KB) | wikimediacommons>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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