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 ${\displaystyle 1\le n\le 2\times 10^9}$. 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 ${\displaystyle 906150257\le n\le 906488079}$. 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

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

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

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