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PreprintCC BY 4.0arXiv

On the lower bound for the L1 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 ℐλ(X)=∫𝕋|∑1leqslant nleqslant Xλ(n)e(nα)| dα ≫ X1/4. This removes the Xε loss in the previous result of Pandey and Radziwiłł~\cite{PandeyRadziwill2026}. In the second paper in this series, we will prove that limsupX → ∞ (log ℐλ(X))/(log X) geqslant1/3.

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Cite

BibTeXRIS
BibTeX
@misc{li2026lower,
  title = {{On the lower bound for the $L^1$ norm of Liouville exponential sums. I}},
  author = {Jiangpeng Li and Yu-Chen Sun and Hui Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03589},
  eprint = {2611.03589},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03589},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the lower bound for the L^1 norm of Liouville exponential sums. I
AU  - Jiangpeng Li
AU  - Yu-Chen Sun
AU  - Hui Wang
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03589
UR  - https://arxiv.org/abs/2611.03589
AB  - Let λ be the Liouville function. We prove Let λ be the Liouville function. We prove ℐ_λ(X)=∫_𝕋|∑_1leqslant nleqslant Xλ(n)e(nα)| dα ≫ X^1/4. This removes the X^ε loss in the previous result of Pandey and Radziwiłł~\cite{PandeyRadziwill2026}. In the second paper in this series, we will prove that limsup_X → ∞ (log ℐ_λ(X))/(log X) geqslant1/3.
ER  -