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arXiv · 2611.03589

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

Jiangpeng Li·Yu-Chen Sun·Hui Wang
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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 \[ \limsup_{X \to \infty} \frac{\log \mathcal{I}_λ(X)} {\log X} \geqslant\frac{1}{3}. \]

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Jiangpeng Li, Yu-Chen Sun, Hui Wang. 2026-11-02. On the lower bound for the $L^1$ norm of Liouville exponential sums. I. https://arxiv.org/abs/2611.03589

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