等差数列上默比乌斯函数的有界和
Bounding sums of the Möbius function over arithmetic progressions
摘要：
设$M(x)=\sum_{1\le n\le x}μ(n)$，其中$μ$是默比乌斯函数。众所周知，黎曼猜想等价于断言$M(x)=O(x^{1/2+ε})$对所有$ε>0$。在黎曼假设的假设下，对$M(x)$的进一步界定已经有了很大的兴趣和进展。在2009年，Soundararajan建立了\[ M(x)\ll\sqrt{x}\exp\left((\log x)^{1/2}(\log\log x)^c\right) \]的当前最佳边界（将$c$设置为$14$，尽管可以减少）。Halupczok和Suger最近应用Soundararajan的方法绑定更一般的总和默比乌斯函数的等差数列，形式\[ M(x;q,a)=\sum_{\substack{n\le x \\ n\equiv a\pmod{q}}}μ(n). \]他们能够表明，假设广义黎曼假设，$M(x;q,a)$满足\[ M(x;q,a)\ll_ε\sqrt{x}\exp\left((\log x)^{3/5}(\log\log x)^{16/5+ε}\right) \]的所有$q\le\exp\left(\frac{\log 2}2\lfloor(\log x)^{3/5}(\log\log x)^{11/5}\rfloor\right)$，与$a$这样，$(a,q)=1$，和$ε>0$。本文改进了Halupczok和Suger的工作，得到了与Soundararajan给出的$M(x)$（$\log x$的指数中有一个$1/2$）的界相同的$M(x;q,a)$的界，并且对模$q$和剩余$a$没有大小或整除性限制.
英文摘要：
Let $M(x)=\sum_{1\le n\le x}μ(n)$ where $μ$ is the Möbius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+ε})$ for all $ε>0$. There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of \[ M(x)\ll\sqrt{x}\exp\left((\log x)^{1/2}(\log\log x)^c\right) \] (setting $c$ to $14$, though this can be reduced). Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the Möbius function over arithmetic progressions, of the form \[ M(x;q,a)=\sum_{\substack{n\le x \\ n\equiv a\pmod{q}}}μ(n). \] They were able to show that assuming the Generalized Riemann Hypothesis, $M(x;q,a)$ satisfies \[ M(x;q,a)\ll_ε\sqrt{x}\exp\left((\log x)^{3/5}(\log\log x)^{16/5+ε}\right) \] for all $q\le\exp\left(\frac{\log 2}2\lfloor(\log x)^{3/5}(\log\log x)^{11/5}\rfloor\right)$, with $a$ such that $(a,q)=1$, and $ε>0$. In this paper, we improve Halupczok and Suger's work to obtain the same bound for $M(x;q,a)$ as Soundararajan's bound for $M(x)$ (with a $1/2$ in the exponent of $\log x$), with no size or divisibility restriction on the modulus $q$ and residue $a$.
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。
