Logarithmic Chowla Correlations Across All Shift Scales
Abstract
Let λ(n)=(−1)Ω(n) be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute c>0 such that every sufficiently large x admits a single set ℰx⊆[1,x] with |ℰx∩[1,H]|≪A H(log x)−A (1≤ H≤ x) for every fixed A>0, while maxsubstack1≤ h≤ x h∉ℰxsup1≤ y≤ x|∑n≤ y(λ(n)λ(n+h))/n|≪(log x)1−c. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of log x. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet L-functions, we also prove, uniformly for h∈ℕ and 1≤ y≤ x, |∑n≤ y(λ(n)λ(n+h))/n|≤log(2minh,y)+O((log x)1−cG) for an absolute cG>0, with no exceptional shifts.
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BibTeX
@misc{guo2026logarithmic,
title = {{Logarithmic Chowla Correlations Across All Shift Scales}},
author = {Jizhou Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.23500},
eprint = {2608.23500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23500},
}
RIS
TY - UNPB
TI - Logarithmic Chowla Correlations Across All Shift Scales
AU - Jizhou Guo
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.23500
UR - https://arxiv.org/abs/2608.23500
AB - Let λ(n)=(−1)^Ω(n) be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute c>0 such that every sufficiently large x admits a single set ℰ_x⊆[1,x] with |ℰ_x∩[1,H]|≪_A H(log x)^−A (1≤ H≤ x) for every fixed A>0, while max_substack1≤ h≤ x h∉ℰ_xsup_1≤ y≤ x|∑_n≤ y(λ(n)λ(n+h))/n|≪(log x)^1−c. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of log x. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet L-functions, we also prove, uniformly for h∈ℕ and 1≤ y≤ x, |∑_n≤ y(λ(n)λ(n+h))/n|≤log(2minh,y)+O((log x)^1−c_G) for an absolute c_G>0, with no exceptional shifts.
ER -