Equidistributions of Sign Patterns of the Liouville Function and Normal Numbers

The equidistribution of the double sign patterns of the Liouville function $\lambda$ is proved unconditionally. As application, it is shown that the computable real number $$\sum _{n\ge 1}\frac{1+\lambda (n)}{2^n}$$ is a simply normal number in base 4.