File:Divisor-distribution.jpeg

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

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

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

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

and ${\displaystyle d(n)}$ is the divisor function and ${\displaystyle \gamma =0.577\dots }$ 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 ${\displaystyle \mathrm{\Delta }(x)}$ 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

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