File:Liouville-log.svg

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Liouville-log.svg‎ (SVG file, nominally 600 × 480 pixels, file size: 26 KB)

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[edit] Summary

Description Logarithmic-scale graph of the summatory en:Liouville function. That is, this is a graph of
L(n)=\sum_{k=1}^n \lambda(k)

for 1\le n \le 2\times 10^9. The summatory function is graphed in red. The green spike indicates the location where the en:Pólya conjecture fails. The Pólya conjecture fails to hold for most values of n in the region of 906150257 \le n\le 906488079. In this region, the function reaches a maximum value of 829 at n=906316571, which is represented by the height of the green bar.

The blue line indicates the contribution of the first non-trivial zero of the Riemann zeta function to the summatory Liouville function. Specifically, the blue line shows

a(n)=\sqrt{n} \,\max\left(0.3, \left|\sin\left( \frac{14.13472514\,\log(n)}{2}+\frac{5\pi}{8} \right)\right|\right)

Here, \vert\cdot\vert denotes the en:absolute value. The only reason for taking the max of 0.3 or the sine function is to avoid cluttering the image. The first non-trivial zero of the Riemann zeta function is located at

s=\frac{1}{2}+i14.13472514\cdots
The other zeroes also contribute to the summatory Liouville function as well; but it is clear that the dominant contribution to the oscillations is from the first zero[1]
Date 4 June 2007(2007-06-04) (original upload date)
Source Original uploaded on en.wikipedia (transferred to commons by Hagman)
Author Created by Linas Vepstas en:User:Linas

[edit] References

  1. ↑ Philippe Flajolet, Linas Vepstas, "On Differences of Zeta Values", arXiv math/0611332

[edit] Licensing

Linas at the English language Wikipedia, the copyright holder of this work, hereby publishes it under the following license:
GNU head 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.

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attribution share alike		 This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Attribution: Linas at the English language Wikipedia
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This licensing tag was added to this file as part of the GFDL licensing update.

[edit] Original upload log

This file was originally uploaded at en.wikipedia as Liouville-log.svg, before it was transfered to Commons.

Upload date | User | Bytes | Dimensions | Comment

  • 2007-06-04 02:49:18 | Linas | 27093 | 600×480 | Created by Linas Vepstas [[User:Linas]] 3 June 2007

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current10:42, 15 January 2011Thumbnail for version as of 10:42, 15 January 2011600 × 480 (26 KB)Hagman{{BotMoveToCommons|en.wikipedia|year={{subst:CURRENTYEAR}}|month={{subst:CURRENTMONTHNAME}}|day={{subst:CURRENTDAY}}}} The upload bot is User:CommonsHelper2 Bot which is called by [http://toolserver.org/~commonshelper2/index.php CommonsHelper2]. The t
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This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.
Width 600
Height 480
Image title Produced by GNUPLOT 4.0 patchlevel 0
Retrieved from "http://en.wikipedia.org/wiki/File:Liouville-log.svg"
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