Improved bounds for the Fourier uniformity conjecture

Cédric Pilatte

Let $\lambda$ denote the Liouville function. We prove that $$\sum _{X\le x<2X}\underset{\alpha \in \mathbb{R}/\mathbb{Z}}{\sup }|\sum _{x\le n<x+H}\lambda (n)e(n\alpha )|=o(HX)$$ as…