Image:Liouville-log.svg

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Liouville-log.svg‎ (SVG file, nominally 600 × 480 pixels, file size: 26 KB)

[edit] Summary

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

$L(n)=\sum_{k=1}^n \lambda(k)$

for $1\le n \le 2\times 10^9$ . The summatory function is graphed in red. The green spike indicates the location where the Pólya conjecture fails. The Pólya conjecture fails to hold for most values of $n$ in the region of $906150257 \le n\le 906488079$ . In this region, the function reaches a maximum value of 829 at $n = 906316571$ , 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

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

Here, $\vert\cdot\vert$ denotes the 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

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

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]

[edit] References

^ Philippe Flajolet, Linas Vepstas, "On Differences of Zeta Values", arXiv math/0611332

[edit] Licensing

Created by Linas Vepstas User:Linas 3 June 2007

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.