Quantitative Logarithmic Chowla Correlations Uniformly over Growing Shifts
2026-08-24 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study the Liouville function, a sequence related to prime factorization, and its correlations at shifted points. Building on previous work by Pilatte, Tao, and Teräväinen, they provide an explicit quantitative bound for the logarithmic average of these correlations over all scales, with shifts growing slowly as a power of the logarithm. Their main advance is a direct estimate valid for shifts up to about (log x)^(1/700), improving understanding of how these correlations behave across different scales. They also discuss technical tools such as a circle-method adaptation and residue-uniform transfer related to certain operators, but their result does not resolve the full Chowla conjecture.
Liouville functiontwo-point correlationlogarithmic averagecircle methodexponential sumsChowla conjectureshort sumsdyadic cutoffnon-backtracking operatorlogarithmic scaling
Authors
Jizhou Guo
Abstract
Let $λ(n)=(-1)^{Ω(n)}$ be the Liouville function. Pilatte proved a fixed power saving for the logarithmically weighted two-point correlation at shift one. More recently, Tao and Teräväinen obtained power-logarithmic two-point estimates uniform over polylogarithmically growing shifts and coefficients outside a common exceptional set of scales; their result in particular implies, after logarithmic integration, a growing-shift logarithmic estimate with some unspecified positive exponent. We give a direct all-scales logarithmic estimate with an explicit shift range. For every fixed $0<κ<1/700$, there are constants $c_κ>0$ and $x_0(κ)$ such that $\displaystyle \sup_{1\le h\le(\log x)^κ}\left|\sum_{n\le x}\frac{λ(n)λ(n+h)}{n}\right|\ll_κ(\log x)^{1-c_κ}\qquad (x\ge x_0(κ)).$ The explicit endpoint is inherited from the $(\log N)^{-1/700}$ term in the short exponential-sum estimate of Matomäki, Radziwiłł and Tao. The key quantitative step is a scale-flexible version of Pilatte's circle-method uncentring: dilation by $h$ preserves the relevant fourth moment, while the short sums cost $h^{1/5}$. A flexible dyadic cutoff recovers every $κ<1/700$. For completeness we also record, in the specialised notation needed here, a residue-uniform arbitrary-interval transfer for the centred non-backtracking operator; a more general decoupling statement appears in the work of Tao and Teräväinen. The result is logarithmically weighted and does not prove the ordinary Cesàro two-point Chowla conjecture.