On the lower bound for the $L^1$ norm of Liouville exponential sums. I

Abstract

Let $λ$ be the Liouville function. We prove Let $λ$ be the Liouville function. We prove \[ \mathcal{I}_λ(X)=\int_{\mathbb{T}}\left|\sum_{1\leqslant n\leqslant X}λ\left(n\right)\mathrm{e}\left(nα\right)\right| \mathrm{d}α \gg X^{\frac{1}{4}}. \] This removes the $X^\varepsilon$ loss in the previous result of Pandey and Radziwiłł~\cite{PandeyRadziwill2026}. In the second paper in this series, we will prove that \[ \limsup_{X \to \infty} \frac{\log \mathcal{I}_λ(X)} {\log X} \geqslant\frac{1}{3}. \]