发表机构
Universidad Autónoma de Madrid; École Polytechnique Fédérale de Lausanne (EPFL)（马德里自治大学; 洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出新方法研究短区间内算术函数与高度无理频率线性相位的相关性，改进了相关定理的指数 barrier，还指出特征扭曲狄利克雷多项式大值估计的改进可带来对应下界的提升。
AI 中文摘要
我们引入一种新方法，用于研究短区间内算术函数与高度无理频率线性相位的相关性。作为应用，我们证明当$x^{0.58+\varepsilon }\leq H\leq x$时，有\nthe absolute value of the sum from n=x+1 to x+H of μ(n) times e(nα) is much less than H times (log x) raised to the power of -1/3+ε。在$H$的相同范围内，我们还证明，对于任意大的$A>0$，除非存在整数$1\leq q\ll (\log x)^{O_A(1)}$使得$\|q\alpha \| \ll x (\log x)^{O_A(1)}/H^2$，否则有\nthe absolute value of the sum from n=x+1 to x+H of Λ(n) times e(nα) is much less than H times (log x) raised to the power of -A。这突破了arXiv:1911.09076v2中定理1.5以及[T. Zhan，《关于大奇数表示为三个几乎相等素数之和》，《数学学报》7.3（1991），259-272]中定理2的$3/5$ barrier。此外，通过我们的方法，特征扭曲狄利克雷多项式大值估计的任何改进都将带来$H$下界的对应提升。
英文摘要
We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequency over short intervals. As an application, we prove that \[ \Big|\sum_{x<n \leq x+H}μ(n) e(nα)\Big| \ll H(\log x)^{-1/3+\varepsilon } \] whenever $x^{0.58+\varepsilon }\leq H\leq x$. In the same range of $H$ we also show that \[ \Big|\sum_{x<n \leq x+H} Λ(n) e(nα)\Big| \ll H(\log x)^{-A} \] for arbitrarily large $A>0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\|qα\| \ll x (\log x)^{O_A(1)}/H^2$. This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials would lead to a corresponding improvement in the lower bound for $H$.
Comments44 pages
