The Liouville function has super-linear block growth

You need to know: Prime factorisation of positive integers, limits.

Background: For a positive integer n, let $\Omega(n)$ be the number of prime factors of n, counted with multiplicity. Function $\lambda(n)=(-1)^{\Omega (n)}$ is known as the Liouville function. The block complexity $P_{\lambda}(n)$ of $\lambda(n)$ is the number of sign patterns of size n that are taken by consecutive values of $\lambda(n)$ .

The Theorem: On 2nd August 2017, Nikos Frantzikinakis and Bernard Host submitted to arxiv a paper in which they proved, among other results, that $\lim\limits_{n\to\infty}\frac{P_{\lambda}(n)}{n}=\infty$ .

Short context: The Chowla conjecture predicts, as one may naturally expect, that all possible sign patterns of size n are taken by the Liouville function. In other words, $P_{\lambda}(n)$ is conjectured to be $2^n$ . However, the best lower bound for $P_{\lambda}(n)$ before 2017 was only $P_{\lambda}(n)\geq n+5$ for $n\geq 3$ . The Theorem significantly improves this bound.