Image:ChebyshevPsi.png

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ChebyshevPsi.png‎ (499 × 308 pixel, file size: 7 KB, MIME type: image/png)

[edit] Summary

Plot of Chebyshev's $ψ$ function, along with the approximation $x - log2π$ . This function can be defined in Mathematica by:

ψ[n_] := Sum[If[PrimeQ[p], Floor[Log[n]/Log[p]] Log[p], 0], {p, 2, n}]
SetAttributes[ψ, Listable]

The above plot was then generated by:

 Plot[{ψ[n], n - Log[2π]}, {n, 0, 50}, 
   AxesLabel -> {n, "ψ(n)"},
   TextStyle -> {FontFamily -> "Times", FontSize -> 16},
   PlotStyle -> {{Hue[0.6], Thickness[0.006]}, {RGBColor[0, 0, 0]}}]

[edit] Licensing

I, the creator of this work, hereby release it into the public domain. This applies worldwide.
In case this is not legally possible,
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(del) (cur) 17:25, 2 July 2006 . . Dantheox (Talk | contribs) . . 499×308 (6,860 bytes) (Plot of Chebyshev's <math>\psi</math> function, along with the approximation <math>x-\log 2\pi</math>. This function can be defined in Mathematica by: ψ[n_] := Sum[If[PrimeQ[p], Floor[Log[n]/Log[p]] Log[p], 0], {p, 2, n}] SetAttributes[ψ, Listabl)