Image:Divisor-summatory-big.svg

From Wikipedia, the free encyclopedia

Image
File history
File links

No higher resolution available.

Divisor-summatory-big.svg‎ (600 × 480 pixel, file size: 35 KB, MIME type: image/svg+xml)

[edit] Divisor summatory function

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

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. Here, $\gamma=0.577\ldots$ is the Euler-Mascheroni constant. The graph extends up to $x = 10 7$ . The green lines that do not quite bound the picture are a graph of $\pm 2x^{7/22}$ and give a general scale for the rate of growth for this function. Curiously, note that the function is not centered on the zero axis, but seems to be closer to the upper curve. Although the graph appears to be noisy, notice that it is less noisy than the corresponding graph for smaller n, and is more noisy than a similar graph for larger n.

[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.

File history

Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version.
Click on date to download the file or see the image uploaded on that date.

(del) (cur) 17:04, 13 July 2006 . . Linas (Talk | contribs) . . 600×480 (35,465 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 )