Improved bounds for the two-point logarithmic Chowla conjecture
Cédric Pilatte
TL;DR
The paper proves a strengthened bound for the two-point logarithmic Chowla correlation of the Liouville function, showing that the logarithmic average of λ(n)λ(n+1) can be bounded by (log x)^{1−c} for some absolute c>0, under current techniques. Central to the approach is recasting the problem into a spectral framework via a weighted adjacency matrix A_Y and its non-backtracking counterpart M_Y, then exploiting the Ihara-Bass formula to relate their spectra. A delicate high-trace argument for the non-backtracking operator is developed, with a detailed combinatorial decomposition of walks into single, lit, and unlit indices, and a dichotomy into predictable vs unpredictable words, enabling the extraction of large triangular systems of divisibility constraints and the application of a combinatorial sieve to handle prohibited progressions. The combination of localization, spectral tail control, and sieve machinery yields the targeted bound and hints at structural generality to broader classes of multiplicative functions, while also informing almost-all-scale bounds for unweighted two-point correlations. The work extends the Helfgott–Radziwiłł program by integrating non-backtracking spectral methods with refined combinatorics, offering a robust framework for future refinements in higher-order correlations and related multiplicative problems.
Abstract
Let $λ$ be the Liouville function, defined as $λ(n) := (-1)^{Ω(n)}$ where $Ω(n)$ is the number of prime factors of $n$ with multiplicity. In 2021, Helfgott and Radziwiłł proved that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll \frac{\log x}{(\log \log x)^{1/2}},$$improving earlier results by Tao and Teräväinen. We prove that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll (\log x)^{1-c}$$for some absolute constant $c>0$. This appears to be best possible with current methods.