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| View Original: | Prime number theorem ratio convergence.svg (250x160) | |||
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| Keywords: Prime number theorem ratio convergence.svg en A plot showing how two estimates described by the prime number theorem <math>\frac x \ln x </math> and <math>\int_2 x \frac 1 \ln t \mathrm d t Li x li x - li 2 </math> converge asymptotically towards <math>\pi x </math> the number of primes less than x The x axis is <math>x</math> and is logarithmic labelled in evenly spaced powers of 10 going up to 10<sup>24</sup> the largest <math>x</math> for which <math>\pi x </math> is currently known The former estimate converges extremely slowly while the latter has visually converged on this plot by 10<sup>8</sup> Source used to generate this chart is shown below own User Dcoetzee 2013-03-21 Prime numbers Cc-zero Source All source released under CC0 waiver Mathematica source to generate graph which was then saved as SVG from Mathematica <pre> Sample both functions at 600 logarithmically spaced points between \ 1 and 2 40 base NE 24 Log10/600 ; ratios Table Roundbase x NPrimePiRoundbase x/ base x/ x Logbase x 1 Floor40/Log2 base ; ratiosli Table Roundbase x NPrimePi Roundbase x/ LogIntegralbase x - LogIntegral2 x CeilingLogbase 2 Floor40/Log2 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 Joinratios Map 1 N 2/ 1/ Log 1 LargePiPrime; ratiosli2 Joinratiosli Map 1 N 2/ LogIntegral 1 - LogIntegral2 LargePiPrime; Plot with log x axis together with the horizontal line y 1 ShowLogLinearPlot1 x 1 10 24 PlotRange -> 0 8 1 25 ListLogLinearPlot ratios2 ratiosli2 Joined -> True LabelStyle -> FontSize -> 14 </pre> LaTeX source for labels <pre> \left \pi x \middle/ \frac x \ln x \right \left \pi x \middle/ \int_2 x \frac 1 \ln t \mathrm d t \right </pre> These were converted to SVG with http //www tlhiv org/ltxpreview/ and then the graph was embedded into the resulting document in Inkscape Axis fonts were also converted to Liberation Serif in Inkscape | ||||
