Let $\lambda$ be the Liouville function; that is, the completely multiplicative function defined by $\lambda(p)=-1$ at every prime $p$. Up to a certain extent, the statistical properties of $\lambda$ are believed to resemble those of a random $\pm 1$ sequence. For example, the Riemann Hypothesis is equivalent to the partial sums $\sum_{n\leq x} \lambda(n)$ exhibiting near square-root cancellation.

In this talk, we consider the pseudo-random behaviour of the Liouville function in short intervals. We will discuss recent work on the Fourier uniformity conjecture, which predicts that the Liouville function has negligible correlations with linear phases $n \mapsto e(n\alpha)$ on almost all intervals of the form $[x, x+H(X)]$ with $x\in [X, 2X]$, where $H(X)$ is a function tending to infinity arbitrarily slowly.