Sign patterns of the Möbius function

I test some of the statements in the paper "Sign patterns of the Liouville and Möbius function". On page 6 in the paper one can find "the three events $\mu(n) = +1, \mu(n) = 0, \mu(n) = -1$ occur with asymptotic probability $\frac{1}{2 \zeta(2)}, 1 - \frac{1}{\zeta(2)}, \frac{1}{2 \zeta(2)}$ respectively". Here, $\mu$ is the Möbius function and $\zeta$ is Riemann's zeta function. I interpret the concept of asymptotic probability in a naive way and just count these events in Mathematica for $n \le 100$.

Here is the code that I used

The code produces the plot below

In this plot, the gray rectangles are the "experimental data" from counting events; the red lines are the exact values. The data thus agrees well with the exact values.

Also on page 6 in the paper, one can find that $(\mu(n), \mu(n+1) )$ takes the value $(0,0)$ with asymptotic probability $$1 - \frac{2}{\zeta(2)} + c = 0.11$$ and each of the four values $(+1,0), (-1,0), (0,+1), (0,-1)$ with asymptotic probability $$\frac{1}{2} \left(\frac{1}{\zeta(2)} - c\right) = 0.14$$ with $c = 0.32$. The paper has a wrong factor of $1/4$ instead of $1/2$. Again, I calculate events for $n \le 10000$. Here is the code I used in this case

The picture produced is this

Again good agreement is found for the cases for which the paper gives exact values.

Related

• blog Terence Tao about the paper