Quantitative Logarithmic Chowla Correlations Uniformly over Growing Shifts

2026-08-24

AI Summary

该研究解决了Liouville函数的对数加权两点相关性估计问题，通过改进Pilatte的方法，在所有尺度上直接给出了具有明确移位范围的估计。

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.

Problem

• Logarithmic Chowla Conjecture
• Liouville function
• logarithmically weighted two-point correlation

Innovation

• logarithmic estimate
• shift range
• circle-method uncentring
• fourth moment