Distribution of neighboring values of the Liouville and Möbius functions

Abstract

Let $\lambda (n)$ and $\mu (n)$ denote the Liouville function and the Möbius function, respectively. In this study, relationships between the values of $\lambda (n)$ and $\lambda (n+h)$ up to $n\le 10^8$ for $1\le h\le 1,000$ are explored. Chowla's conjecture predicts that the conditional expectation of $\lambda (n+h)$ given $\lambda (n)=1$ for $1\le n\le X$ converges to the conditional expectation of $\lambda (n+h)$ given $\lambda (n)=-1$ for $1\le n\le X$ as $X\to \infty$. However, for finite $X$, these conditional expectations are different. The observed difference, together with the significant difference in ${\chi }^2$ tests of independence, reveals hidden additive properties among the values of the Liouville function. Similarly, such additive structures for $\mu (n)$ for square-free $n$'s are identified. These findings pave the way for developing possible, and hopefully efficient, additive algorithms for these functions. The potential existence of fast, additive algorithms for $\lambda (n)$ and $\mu (n)$ may eventually provide scientific evidence supporting the belief that prime factorization of large integers should not be too difficult. For $1\le h\le 1,000$, the study also tested the convergence speeds of Chowla's conjecture and found no relation on $h$.