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Summary

DescriptionPrime number theorem absolute error.svg
English: A log-log plot showing the absolute error of two estimates to the prime-counting function π ( x ) {\displaystyle \pi (x)} {\displaystyle \pi (x)}, given by x ln ⁡ x {\displaystyle {\frac {x}{\ln x}}} {\displaystyle {\frac {x}{\ln x}}} and ∫ 2 x 1 ln ⁡ t d t = L i ( x ) = l i ( x ) − l i ( 2 ) {\displaystyle \int _{2}^{x}{\frac {1}{\ln t}}\mathrm {d} t=Li(x)=li(x)-li(2)} {\displaystyle \int _{2}^{x}{\frac {1}{\ln t}}\mathrm {d} t=Li(x)=li(x)-li(2)}. The x axis is x {\displaystyle x} {\displaystyle x} and is logarithmic (labelled in evenly spaced powers of 10), going up to 1024, the largest x {\displaystyle x} {\displaystyle x} for which π ( x ) {\displaystyle \pi (x)} {\displaystyle \pi (x)} is currently known. The y axis is also logarithmic, going up to the absolute error of x / ln ⁡ x {\displaystyle x/\ln x} {\displaystyle x/\ln x} at 1024. The error of both functions appears to increase as a power of x {\displaystyle x} {\displaystyle x}, 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 π ( x ) {\displaystyle \pi (x)} {\displaystyle \pi (x)} in the larger region. Source used to generate this chart is shown below.
Date
Source Own work
Author Dcoetzee
SVG development
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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.
Source code
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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

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

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse

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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current14:47, 21 March 2013 Thumbnail for version as of 14:47, 21 March 2013 283 ×ばつ 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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