# File:Mangoldt-series.svg

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Original file(SVG file, nominally 600 × 480 pixels, file size: 13 KB)

## Summary

 Description English: Graph of a series involving the von Mangoldt function. More precisely, this is a graph of the function ${\displaystyle F(y)=\sum _{n=2}^{\infty }\left(\Lambda (n)-1\right)e^{-ny}}$ considered by Hardy and Littlewood in 1916. They demonstrated that ${\displaystyle F(y)={\mathcal {O}}\left({\sqrt {\frac {1}{y}}}\right)}$ Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there exists a value ${\displaystyle K>0}$ such that ${\displaystyle F(y)<-{\frac {K}{\sqrt {y}}}}$ and ${\displaystyle F(y)>{\frac {K}{\sqrt {y}}}}$ 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 ${\displaystyle 10^{-5} 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 ${\displaystyle y=0}$. For the oscillations shown in this graph, a summation including more than 2 billion terms of the series was required. Date 3 July 2006; 4 July 2006 (original upload date) Source Own work Author Linas Vepstas (User:Linas)

## Licensing

 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. A copy of the license is included in the section entitled GNU Free Documentation License.
 This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one. This licensing tag was added to this file as part of the GFDL licensing update.

## Original upload log

Transferred from en.wikipedia to Commons using For the Common Good.

The original description page was here. All following user names refer to en.wikipedia.
Date/Time Dimensions User Comment
15:19, 4 July 2006 600 × 480 (12,300 bytes) w:en:Linas (talk | contribs) higher precision, 2 billion terms
04:40, 4 July 2006 600 × 480 (9,078 bytes) w:en:Linas (talk | contribs) oops, its upside-down
04:14, 4 July 2006 600 × 480 (9,080 bytes) w:en:Linas (talk | contribs) Graph of a series involving the [[von Mangoldt function]]. Created by Linas Vepstas [[User:Linas]] on 3 July 2006

## File history

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Date/TimeThumbnailDimensionsUserComment
current17:40, 25 October 2015600 × 480 (13 KB)Pqnlrn (talk | contribs)Reverted to version as of 18:30, 2 November 2013 (UTC)
17:39, 25 October 2015600 × 420 (49 KB)Pqnlrn (talk | contribs)Rewrote the ordinate axis label in graphical form
18:30, 2 November 2013600 × 480 (13 KB)McPot (talk | contribs)Trying to fix broken SVG-thumbnail. [inkscape --export-plain-svg] (Uploaded new version of file using Wiki.java)
15:13, 16 October 2013No thumbnail0 × 0 (12 KB)OgreBot (talk | contribs)(BOT): Reverting to most recent version before archival
15:12, 16 October 2013No thumbnail0 × 0 (9 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2006-07-04 04:40:04 by Linas
15:12, 16 October 2013No thumbnail0 × 0 (9 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2006-07-04 04:14:52 by Linas
16:04, 3 September 2013No thumbnail0 × 0 (12 KB)Clarkcj12 (talk | contribs)Transferred from en.wikipedia: see original upload log above
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