Semicircle law (Quantum Hall)

Summary

Graph of a series involving the von Mangoldt function. More precisely, this is a graph of the function

$F(y)=\sum _{n=2}^{\mathrm{\infty }}(\mathrm{\Lambda }(n)-1)e^{-ny}$

considered by Hardy and Littlewood in 1916. They demonstrated that

$F(y)=\mathcal{𝒪}\left(\sqrt{\frac{1}{y}}\right)$

Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there exists a value $K>0$ such that

$F(y)<-\frac{K}{\sqrt{y}}$ and $F(y)>\frac{K}{\sqrt{y}}$

infinitely often. This graph demonstrates that the second condition is not immediately apparent, numerically. The graph of this function appears to be remarkably linear in the region $10^{-5}<y<1/2$ and visually appears to have an intercept with the y-axis at about -0.337877. However, on closer examination, one discovers oscillations of increasing magnitude as the function approaches $y=0$. For the oscillations shown in this graph, a summation including more than 2 billion terms of the series was required.

Licensing

Created by Linas Vepstas User:Linas on 3 July 2006 Template:GFDL