增长移位下的定量对数Chowla相关性
Logarithmic Chowla Correlations Across All Shift Scales
浏览论文内容
中文总结 AI 辅助
本文针对增长移位下的刘维尔函数对数加权两点相关性，给出了全尺度的定量估计，明确了移位范围与相关常数，为Chowla猜想的研究提供了新的定量结果。
中文摘要 AI 辅助
设λ(n)=(-1)^(Ω(n))为刘维尔函数。Pilatte证明了移位为1时对数加权两点相关性存在固定幂次节省。最近，Tao和Teräväinen在多对数增长移位及公共例外尺度集外的系数上，得到了幂对数两点估计；其结果经对数积分后，特别蕴含了具有某个未指定正指数的增长移位对数估计。本文给出了直接的全尺度对数估计，具有明确的移位范围：对每个固定的0<κ<1/700，存在常数c_κ>0和x_0(κ)，使得当x≥x_0(κ)时，有sup_{1≤h≤(log x)^κ}|∑_{n≤x}λ(n)λ(n+h)/n|≪_κ (log x)^(1−c_κ)。该明确端点继承自Matomäki、Radziwiłł和Tao的短指数和估计中的(log N)^(−1/700)项。关键定量步骤是Pilatte圆方法非中心化的尺度灵活版本：h的伸缩保持相关四阶矩，而短和的代价为h^(1/5)；灵活二进截断恢复了所有κ<1/700的情形。为完整起见，本文还以所需的专门记号记录了中心化非回溯算子的残差均匀任意区间转移，更一般的解耦陈述见于Tao和Teräväinen的工作。该结果为对数加权的，未证明普通Cesàro两点Chowla猜想。
英文摘要
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 $\mathcal E_x\subseteq[1,x]$ with $|\mathcal E_x\cap[1,H]|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left|\sum_{n\le y}\frac{λ(n)λ(n+h)}{n}\right|\ll(\log x)^{1-c}$. 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 $h\in\mathbb N$ and $1\le y\le x$, $\left|\sum_{n\le y}\frac{λ(n)λ(n+h)}{n}\right|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.
