Image:PlotDelta.gif

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PlotDelta.gif‎ (800 × 480 pixels, file size: 18 KB, MIME type: image/gif)

[edit] Summary

This plot illustrates normalized fluctuations of the prime-counting function π(x) on the logarithmic scale.

The red curve represents delta function, while the blue curve represents similar function where π₀(x) is replaced by li(x), so one can compare the fluctuations with the difference between π₀(x) and li(x).

The image was built by Droog_Andrey using the Mathematica 5.0 Plot function as follows:

\!\(Plot[{\((PrimePi[Exp[u]] - 1 - Sum[\(\(u^k/\(k!\)\)/k\)/Zeta[k + 1], {k, 1, 250}] + 1/u - ArcTan[Pi/u]/Pi)\)*u*Exp[\(-u\)/2], \((ExpIntegralEi[u] - 1 - Sum[\(\(u^k/\(k!\)\)/k\)/Zeta[k + 1], {k, 1, 250}] + 1/u - ArcTan[Pi/u]/Pi)\)*u*Exp[\(-u\)/2]}, {u, 0, 21}, Ticks -> {{{Log[10], l0}, {Log[10]*2, l0\^2}, {Log[10]*3, l0\^3}, {Log[10]*4, l0\^4}, {Log[10]*5, l0\^5}, {Log[10]*6, l0\^6}, {Log[10]*7, l0\^7}, {Log[10]*8, l0\^8}, {Log[10]*9, l0\^9}}, Automatic}, PlotPoints -> 1000, PlotRange -> All, PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}, TextStyle -> {FontSize -> 24}]\)

It is also available on the web: http://primefan.ru:8014/WWW/stuff/primes/plot.gif

[edit] Licensing

I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide. In case this is not legally possible, I grant any entity the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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