PreprintarXiv licencearXiv
Odd order cases of Chowla's averages over arbitrary rectangles
Abstract
Let λ be the Liouville function. In this paper we prove an average form of Chowla Conjecture for even $k $. that is, for every fixed even integer k≥ 2, we prove that limmin{M,N}→ ∞1/MN∑m=1M∑n=1Nλ(m)λ(m+a1n)⋯λ(m+akn)=0. We construct a stationary model for both averaging variables, derive the finite conditional-entropy estimates. A comparison of overlapping intervals on the logarithmic axis then forces this function to vanish.
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BibTeX
@misc{huo2026order,
title = {{Odd order cases of Chowla's averages over arbitrary rectangles}},
author = {Qiang Huo and Zhengxing Lian and Kangbo Ouyang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12869},
eprint = {2610.12869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12869},
}
RIS
TY - UNPB
TI - Odd order cases of Chowla's averages over arbitrary rectangles
AU - Qiang Huo
AU - Zhengxing Lian
AU - Kangbo Ouyang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12869
UR - https://arxiv.org/abs/2610.12869
AB - Let λ be the Liouville function. In this paper we prove an average form of Chowla Conjecture for even $k $. that is, for every fixed even integer k≥ 2, we prove that lim_min{M,N}→ ∞1/MN∑_m=1^M∑_n=1^Nλ(m)λ(m+a_1n)⋯λ(m+a_kn)=0. We construct a stationary model for both averaging variables, derive the finite conditional-entropy estimates. A comparison of overlapping intervals on the logarithmic axis then forces this function to vanish.
ER -