AI 中文总结
研究由莫比乌斯函数扭曲的对角埃利奥特 - 哈尔伯施塔姆问题的加权平均变体，在广义黎曼假设等条件下，用索伯列夫空间或赫尔德 - 齐格蒙德空间权重，得到与猜想‘对角版本’界一致的结果，索伯列夫权重适用范围更广，赫尔德 - 齐格蒙德权重有特定\(\theta\)范围。
AI 中文摘要
回顾由莫比乌斯函数\(\mu(n)\)扭曲的所谓埃利奥特 - 哈尔伯施塔姆猜想，即对于每个\(A > 0\)，有\(\sum_{q\leq N^{\theta}}\max_{y\leq N}\max_{(a,q)=1}\left|\sum_{\underset{\scriptstyle n\equiv a\,\mod\,q}{n\leq y}}\Lambda(n)\mu\left(N - n\right)-\frac{1}{\varphi\left(q\right)}\sum_{n\leq y}\Lambda(n)\mu\left(N - n\right)\right|\ll\frac{N}{\log\left(N\right)^{A}}\)，其中\(0 < \theta < 1\)固定，且该猜想与经典埃利奥特 - 哈尔伯施塔姆猜想结合可证明二元哥德巴赫猜想。本文研究此问题的加权平均变体。在广义黎曼假设、戈内克 - 赫伊哈尔猜想的弱版本下，使用索伯列夫空间\(W^{2,1}\)或赫尔德 - 齐格蒙德空间\(\mathcal{C}^{\delta}\)中合适范围的权重，平均的界与该猜想‘对角版本’的界一致。特别地，索伯列夫空间权重时，对整个\(0 < \theta < 1\)成立一致上界；赫尔德 - 齐格蒙德类\(\mathcal{C}^{\delta}\)权重时，\(\theta\)取决于\(\delta\)选择但仍不低于\(1/2 - 2\varepsilon\)阈值。
英文摘要
Recalling that the so-called Elliott-Halberstam conjecture twisted by the Möbius function $μ(n)$ claims that \[ \sum_{q\leq N^θ}\max_{y\leq N}\max_{(a,q)=1}\left|\sum_{\underset{\scriptstyle n\equiv a\,\mod\,q}{n\leq y}}Λ(n)μ\left(N-n\right)-\frac{1}{φ\left(q\right)}\sum_{n\leq y}Λ(n)μ\left(N-n\right)\right|\ll\frac{N}{\log\left(N\right)^{A}} \] for every $A>0$, where $0<θ<1$ is fixed, and also recalling that the validity of this conjecture, in combination with the validity of the classical Elliott-Halberstam for suitable $θ$, proves the binary Goldbach conjecture, in this paper we study weighted average variants of this problem. We will show that, under Generalized Riemann Hypothesis, a weak version of the Gonek-Hejhal conjecture and working with weights belonging to the Sobolev space $W^{2,1}$ or in the Hölder-Zygmund spaces $\mathcal{C}^δ$ for suitable range of $δ$, the bound of the average is consistent with the bound of the ``diagonal versions'' of this conjecture (that is, taking $y=N$ and taking $n\equiv N\mod q)$. In particular, in the case of weights in Sobolev space, the consistent upper bound holds for the whole $0<θ<1$ and, in the case of weights in the Hölder-Zygmund class $\mathcal{C}^δ$, for $θ$ that depends on the choice of $δ$ but still not below the $1/2-2\varepsilon$ threshold.
