Liouville-log

Autor/Urheber:

Created by Linas Vepstas en:User:Linas

Shortlink:

https://dewiki.de/b/3484fd

Quelle:

Wikimedia Commons

Größe:

600 x 480 Pixel (27093 Bytes)

Beschreibung:

Logarithmic-scale graph of the summatory en:Liouville function. That is, this is a graph of

${\displaystyle L(n)=\sum _{k=1}^{n}\lambda (k)}$

for ${\displaystyle 1\leq n\leq 2\times 10^{9}}$. 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 ${\displaystyle n}$ in the region of ${\displaystyle 906150257\leq n\leq 906488079}$. In this region, the function reaches a maximum value of 829 at ${\displaystyle n=906316571}$, 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

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

Here, ${\displaystyle \vert \cdot \vert }$ 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

${\displaystyle s={\frac {1}{2}}+i14.13472514\cdots }$

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]

Lizenz:

CC BY-SA 3.0

Credit:

Original uploaded on en.wikipedia (transferred to commons by Hagman)

Bild teilen: