Fourier uniformity of the Liouville function in shorter intervals

par Cédric Pilatte (University of Oxford, Royaume-Uni)

lundi 12 janv. 2026, 15:30 → 16:30 Europe/Paris

Salle Yvette Cauchois (IHP - Bâtiment Perrin)

In this talk, we consider the pseudo-random behaviour of the Liouville function in short intervals. Let $H(X)$ be an increasing function. We say that the Liouville function is <em>Fourier-uniform</em> in almost all intervals of length $H = H(X)$ at scale $X$ if $$\sum_{X < x < 2X} \sup_{\alpha \in \mathbb{R}}\, \left\lvert \sum_{x < n < x+H} \lambda(n) e(n\alpha) \right\rvert = o(HX)$$ as $X \to \infty$.
The Fourier uniformity conjecture predicts that this estimate holds for any function $H(X)$ going to infinity with $X$ (arbitrarily slowly). This would generalise the Matomäki-Radziwiłł theorem, which establishes this when $\alpha$ is fixed and independent of $x$.

In 2023, Walsh showed that $\lambda$ is Fourier-uniform in almost all intervals of length $\exp((\log X)^{1/2+\varepsilon})$. We improve this to $\exp((\log X)^{2/5+\varepsilon})$, a natural barrier for this problem.