On the lower bound for the L¹ norm of Liouville exponential sums. I

Let $λ$ be the Liouville function. We prove Let $λ$ be the Liouville function. We prove \[ I_λ(X)=∫T|∑1 n Xλ(n)e(nα)| dα X1/4. \] This removes the X^ε loss in the previous result of Pandey and Radziwiłł~{PandeyRadziwill2026}. In the second paper in this series, we will prove that \[ X → ∞ log I_λ(X)/log X 1/3. \]