English: This plot illustrates normalized fluctuations of the prime-counting function ${\displaystyle \pi (x)}$ on the logarithmic scale.

The red line represents Δ-function, while the blue curve represents similar function where ${\displaystyle \pi _{0}(x)}$ is replaced by ${\displaystyle \mathrm {li} (x)}$, so one can compare the fluctuations with the difference between ${\displaystyle \pi (x)}$ and ${\displaystyle \mathrm {li} (x)}$.

The image was built by Droog_Andrey using Wolfram Mathematica 7.0 as follows:

LogLinearPlot[{(PrimePi[x] - RiemannR[x] + 1/Log[x] - ArcTan[Pi/Log[x]]/Pi)*Log[x]/Sqrt[x], (LogIntegral[x] - RiemannR[x] + 1/Log[x] - ArcTan[Pi/Log[x]]/Pi)*Log[x]/Sqrt[x]}, {x, 1, Exp[14]}, PlotPoints -> 10000, PlotRange -> All, PlotStyle -> {Red, Blue}, AxesOrigin -> {1, 0}, AspectRatio -> 1/2, AxesStyle -> Thick, TicksStyle -> Directive[Thick, Bold, 16]]