素数的高阶一致性与莫比乌斯函数在较短区间内的抵消
Higher order uniformity of the primes and cancellation of the Möbius function in shorter intervals
- University of Turku（图尔库大学）
- Courant Institute of Mathematical Sciences（库朗数学科学研究所）
- Universidad Autónoma de Madrid（马德里自治大学）
- University of Cambridge（剑桥大学）
- École Polytechnique Fédérale de Lausanne (EPFL)（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总结：
本文改进了素数在短区间上的Gowers一致性和莫比乌斯函数在短区间上的抵消，通过将三线性和的均值估计归结为分段线性函数的正性，并融合Guth-Maynard大值估计，得到了更优的指数。
AI中文摘要：
我们证明了von Mangoldt函数与其Cramér模型之差在所有短区间$[X,X+X^{3/5+\varepsilon}]$上的Gowers一致性，改进了第一作者和第四作者与Shao和Tao的工作，其中指数为$5/8$。这也蕴含了素数中线性方程在如此短区间内的局部到整体定理。我们还证明了莫比乌斯函数在所有区间$[X,X+X^{19/35+\varepsilon}]$上具有抵消，改进了第一作者和第四作者的工作，其中指数为$11/20$。两项改进都基于对短区间上三线性和的改进处理，这自然归结为估计三个Dirichlet多项式乘积的均值。在一般设置中，我们将估计具有给定大值界的Dirichlet多项式乘积的均值这一任务归结为证明某个分段线性函数为正。这一归约使得近期的大值估计得以纳入。将该一般框架与源于Guth和Maynard工作的最新大值估计相结合，我们证明了在长度为$X^{19/35+\varepsilon}$的区间上的类型I/II和的Heath-Brown--Iwaniec型估计，以及在长度为$X^{3/5+\varepsilon}$的区间上的一般三线性和的Baker--Harman--Pintz平行四边形型估计。
英文摘要：
We prove the Gowers uniformity of the von Mangoldt function minus its Cramér model in all short intervals $[X,X+X^{3/5+\varepsilon}]$, improving on the work of the first and fourth authors with Shao and Tao, where the exponent was $5/8$. This also implies a local-to-global theorem for linear equations in primes in such short intervals. We also show that the Möbius function has cancellation in all intervals $[X,X+X^{19/35+\varepsilon}]$, improving on the work of the first and fourth authors, where the exponent was $11/20$. Both improvements are based on improved treatment of trilinear sums over short intervals which naturally reduces to estimating mean values of products of three Dirichlet polynomials. In a general setting, we reduce the task of estimating mean values of products of Dirichlet polynomials with given large value bounds to the task of showing that a certain piecewise linear function is positive. This reduction allows recent large value estimates to be incorporated. Combining this general framework with the most recent large value estimates stemming from the work of Guth and Maynard, we prove a Heath-Brown--Iwaniec type estimate for type I/II sums in intervals of length $X^{19/35+\varepsilon}$ and a Baker--Harman--Pintz parallelogram type estimate for general trilinear sums in intervals of length $X^{3/5+\varepsilon}$.
