Image:Divisor-distribution.jpeg

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[edit] Divisor summatory function

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

Δ(x) = D (x) - x log x - x (2γ - 1)

where $D (x)$ is the divisor summatory function

$D(x)=\sum_{n\le x} d(n)$

and $d (n)$ is the divisor function and $\gamma=0.577\ldots$ 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 $x = 0$ to $x = 10 7$ , and so the first $107$ values of $Δ(x)$ are graphed. The y-axis is scaled, so that, from bottom to top, the height of the image is $2 x 7 / 22$ . The line $y = 0$ 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 $\pm x^{7/22}$ 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 $x = 0$ on the left to $x = 10 7$ on the right.

[edit] Licensing

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.

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(del) (cur) 04:17, 14 July 2006 . . Linas (Talk | contribs) . . 500×500 (59,272 bytes) (== Divisor summatory function== This graph illustrates the divisor summatory function with the leading asymptotic terms subtracted. That is, it is a graph of :<math>D(x)-x\log x - x(2\gamma-1)</math> where <math>D(x)</math> is the divisor summatory )