![Graph of a series involving the von Mangoldt function. More precisely, this is a graph of the function F ( y ) = ∑ n = 2 ∞ ( Λ ( n ) − 1 ) e − n y {\displaystyle 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 ) = O ( 1 y ) {\displaystyle 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 {\displaystyle K>0} such that F ( y ) < − K y {\displaystyle F(y)<-{\frac {K}{\sqrt {y}}}} and F ( y ) > K y {\displaystyle 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 {\displaystyle 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 {\displaystyle y=0} . For the oscillations shown in this graph, a summation including more than 2 billion terms of the series was required.](http://upload.wikimedia.org/wikipedia/commons/thumb/9/94/Mangoldt-series.svg/600px-Mangoldt-series.svg.png)
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current | 17:40, 25 October 2015 | ![]() | 600 × 480 (13 KB) | Pqnlrn | Reverted to version as of 18:30, 2 November 2013 (UTC) |
17:39, 25 October 2015 | ![]() | 600 × 420 (49 KB) | Pqnlrn | Rewrote the ordinate axis label in graphical form | |
18:30, 2 November 2013 | ![]() | 600 × 480 (13 KB) | <bdi>McPot</bdi> | Trying to fix broken SVG-thumbnail. [inkscape --export-plain-svg] (Uploaded new version of file using Wiki.java) | |
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15:12, 16 October 2013 | No thumbnail | 0 × 0 (9 KB) | OgreBot | (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2006-07-04 04:14:52 by Linas | |
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