This image illustrates the divisor summatory function with the leading asymptotic terms subtracted. That is, it is a graph of

${\displaystyle \Delta (x)=D(x)-x\log x-x(2\gamma -1)}$

where ${\displaystyle D(x)}$ is the divisor summatory function

${\displaystyle D(x)=\sum _{n\leq x}d(n)}$

and ${\displaystyle d(n)}$ is the divisor function and ${\displaystyle \gamma =0.577\ldots }$ is the Euler-Mascheroni constant.

Properly speaking, the image is of the distribution of the values of the divisor summatory function, with each vertical slice being a histogram. Along the x-axis, x runs from ${\displaystyle x=0}$ to ${\displaystyle x=10^{7}}$, and so the first ${\displaystyle 10^{7}}$ values of ${\displaystyle \Delta (x)}$ are graphed. The y-axis is scaled, so that, from bottom to top, the height of the image is ${\displaystyle 2x^{7/22}}$. The line ${\displaystyle y=0}$ runs horizontally down the center of the image. The histogramming is such that the areas which have a high density of points are colored red, progressively fading out to yellow, green, blue and finally black. Note that the bound ${\displaystyle \pm x^{7/22}}$ is quite tight, and there are many points that actually lie outside this image. However, the image does indicate their relative rarity. In short, this image indicates that although the divisor summatory function is quite random, it does seem to have rather well-behaved statistical properties, and seems to have a narrowing standard deviation as moving from ${\displaystyle x=0}$ on the left to ${\displaystyle x=10^{7}}$ on the right.

Created by Linas Vepstas User:Linas 12 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.

GFDL GNU Free Documentation License //en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License true true

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

This licensing tag was added to this file as part of the GFDL licensing update.

Cc-by-sa-3.0Attribution-ShareAlike 3.0https://creativecommons.org/licenses/by-sa/3.0/

This file is a candidate to be copied to Wikimedia Commons.

Any user may perform this transfer; refer to Wikipedia:Moving files to Commons for details.

If this file has problems with attribution, copyright, or is otherwise ineligible for Commons, then remove this tag and DO NOT transfer it; repeat violators may be blocked from editing.

Other Instructions

• Endorse this file for transfer by adding |human=<your username> to this Template.
• If this file is freely licensed, but otherwise unsuitable for Commons (e.g. out of Commons' scope, still copyrighted in the US), then replace this Template with {{Do not move to Commons|reason=<Why it can't be moved>}}
• If you think that a local copy of this file should be retained, then replace this Template with {{Keep local|reason=<Why a local copy is needed>}}
• Please ensure that the file has a properly descriptive and unambiguous name before transferring; see Wikipedia:File mover#What files should be renamed? for details.

This file was suggested for transfer by a bot (User:Fbot). Please verify that this file is suitable for Commons before transferring it.