Equidistributions of Sign Patterns of the Liouville Function and Normal Numbers

TL;DR

This paper proves the unconditioned equidistribution of Liouville function sign patterns and demonstrates that a related computable number is simply normal in base 4, linking number theory and normality properties.

Contribution

It establishes the unconditioned equidistribution of Liouville sign patterns and shows a specific computable number is simply normal in base 4.

Findings

Liouville function sign patterns are equidistributed unconditionally

A specific computable number is simply normal in base 4

Connects number theory with normality in real numbers

Abstract

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 $$\sum_{n \geq 1} \frac{1+\lambda(n)}{2^{n}}$$ is a simply normal number in base 4.