Image:Mangoldt-series.svg

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[edit] Summary

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

$F(y)=\sum_{n=2}^\infty \left(\Lambda(n)-1\right) e^{-ny}$

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.

[edit] Licensing

Created by Linas Vepstas User:Linas on 3 July 2006

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.

File history

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(del) (cur) 15:19, 4 July 2006 . . Linas (Talk | contribs) . . 600×480 (12,300 bytes) (higher precision, 2 billion terms)
(del) (rev) 04:40, 4 July 2006 . . Linas (Talk | contribs) . . 600×480 (9,078 bytes) (oops, its upside-down)
(del) (rev) 04:14, 4 July 2006 . . Linas (Talk | contribs) . . 600×480 (9,080 bytes) (Graph of a series involving the von Mangoldt function. Created by Linas Vepstas User:Linas on 3 July 2006)