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