File:Prime number theorem ratio convergence.svg
Original file (SVG file, nominally 250 × 160 pixels, file size: 87 KB)
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Summary
[edit]DescriptionPrime number theorem ratio convergence.svg |
English: A plot showing how two estimates described by the prime number theorem, and converge asymptotically towards , the number of primes less than x. 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 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. |
Date | |
Source | Own work |
Author | Dcoetzee |
SVG development InfoField | This chart was created with Mathematica. |
Licensing
[edit]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
[edit]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.
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 13:07, 21 March 2013 | 250 × 160 (87 KB) | Dcoetzee (talk | contribs) | Change n to x to match article | |
12:30, 21 March 2013 | 250 × 160 (86 KB) | Dcoetzee (talk | contribs) | Convert formula from graphics to pure SVG using http://www.tlhiv.org/ltxpreview/ | ||
12:23, 21 March 2013 | 250 × 160 (130 KB) | Dcoetzee (talk | contribs) | {{Information |Description ={{en|1=A plot showing how two estimates described by the prime number theorem, <math>\frac{n}{\ln n}</math> and <math>\int_2^n \frac{1}{\ln t} \mathrm{d}t = Li(n) = li(n) - li(2)</math> converge asymptotically towards <ma... |
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Width | 250.2515 |
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Height | 160.46599 |