扩张、可分性和奇偶性
Expansion, divisibility and parity
摘要：
令$\mathbf{P} \subset [H_0,H]$是素数的集合，其中$\log H_0 \geq (\log H)^{2/3 + ε}$。让$\mathscr{L} = \sum_{p \in \mathbf{P}} 1/p$。令$N$使得$\log H \leq (\log N)^{1/2-ε}$ .证明了存在密度接近于$1$的子集$\mathscr{X} \subset (N, 2N]$，使得线性算子$$(A_{|\mathscr{X}} f)(n) = \sum_{\substack{p \in \mathbf{P} : p | n \\ n, n \pm p \in \mathscr{X}}} f(n \pm p) \; - \sum_{\substack{p \in\mathbf{P} \\ n, n \pm p \in \mathscr{X}}} \frac{f(n \pm p)}{p}$$的所有特征值都是$O(\sqrt{\mathscr{L}})$ .这个界限是最佳的常数因子。换句话说，我们证明了一个图描述整除素数是一个强大的本地扩展几乎无处不在，并确实在一个常数因子的“本地Ramanujan”（即）。专门的$f(n) = λ(n)$与$λ(n)$的刘维函数，并使用Matomäki，Radziwienkirk和陶的平均$λ(n)$在短时间间隔的估计，我们得到的\[\frac{1}{\log x} \sum_{n\leq x} \frac{λ(n) λ(n+1)}{n} = O\Big(\frac{1}{\sqrt{\log \log x}}\Big),\]改进的结果陶的。我们还证明，$\sum_{N<n\leq 2 N} λ(n) λ(n+1)=o(N)$在几乎所有的尺度与类似的误差项，提高陶和Teräväinen的结果。(Tao陶-泰拉维宁采用了不同的方法，基于熵，而不是膨胀;值得注意的是，我们可以取更大的$H$值，从而考虑更多的素数。）我们还可以轻松地证明更清晰的结果。例如：让$S_{N,k}$是所有$N<n\leq 2N$的集合，使得$Ω(n) = k$。然后，对于任何固定的$k$和$k = \log \log N + O(\sqrt{\log \log N})$值（即$k$的任何“流行”值），几乎在所有尺度下，$λ(n+1)$对$S_{N,k}$的平均值都是$o(1)$。
英文摘要：
Let $\mathbf{P} \subset [H_0,H]$ be a set of primes, where $\log H_0 \geq (\log H)^{2/3 + ε}$. Let $\mathscr{L} = \sum_{p \in \mathbf{P}} 1/p$. Let $N$ be such that $\log H \leq (\log N)^{1/2-ε}$. We show there exists a subset $\mathscr{X} \subset (N, 2N]$ of density close to $1$ such that all the eigenvalues of the linear operator $$(A_{|\mathscr{X}} f)(n) = \sum_{\substack{p \in \mathbf{P} : p | n \\ n, n \pm p \in \mathscr{X}}} f(n \pm p) \; - \sum_{\substack{p \in\mathbf{P} \\ n, n \pm p \in \mathscr{X}}} \frac{f(n \pm p)}{p}$$ are $O(\sqrt{\mathscr{L}})$. This bound is optimal up to a constant factor. In other words, we prove that a graph describing divisibility by primes is a strong local expander almost everywhere, and indeed within a constant factor of being "locally Ramanujan" (a.e.). Specializing to $f(n) = λ(n)$ with $λ(n)$ the Liouville function, and using an estimate by Matomäki, Radziwiłł and Tao on the average of $λ(n)$ in short intervals, we derive that \[\frac{1}{\log x} \sum_{n\leq x} \frac{λ(n) λ(n+1)}{n} = O\Big(\frac{1}{\sqrt{\log \log x}}\Big),\] improving on a result of Tao's. We also prove that $\sum_{N<n\leq 2 N} λ(n) λ(n+1)=o(N)$ at almost all scales with a similar error term, improving on a result by Tao and Teräväinen. (Tao and Tao-Teräväinen followed a different approach, based on entropy, not expansion; significantly, we can take a much larger value of $H$, and thus consider many more primes.) We can also prove sharper results with ease. For instance: let $S_{N,k}$ the set of all $N<n\leq 2N$ such that $Ω(n) = k$. Then, for any fixed value of $k$ with $k = \log \log N + O(\sqrt{\log \log N})$ (that is, any "popular" value of $k$) the average of $λ(n+1)$ over $S_{N,k}$ is $o(1)$ at almost all scales.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
