论文标题
功能字段中乘法功能的相关性
Correlations of multiplicative functions in function fields
论文作者
论文摘要
我们开发一种研究性格总和的方法，由乘务函数加权$ f：\ mathbb {f} _q [t] \ to s^1 $，form \ begin {equination} \ sum_ {g \ in \ sum_ {g \ in \ mathcal {m} _n} _n} _n} _n} _n} f（g）$ nise $ nise $ nise $ nise $ nise $ nise us $ nise $ $ξ$是$ \ mathbb {f} _q [t]的简短间隔字符。$我们然后推断出Matomäki-Radziwill定理的版本和Tao的两点差距Elliott Conjocture contienture contienture contients $ \ mathbb {f} _q [t] $，$ q $ q $ quints $ \ mathbb {f} $ q。这些前者在Gorodetsky的工作中有所改善，后者扩展了Sawin-Shusterman在Möbius函数相关性方面的工作，以$ Q $的各种值。 与整数设置相比，我们遇到了不同的现象，特别是在$ Q $的情况下是$ 2 $的情况下的一个低特征问题。 作为我们结果的应用，我们简短地证明了Kátai的猜想的功能字段版本，以较小的增量分类，并获得了分类，并且证明与整数情况不同。 在伴侣论文中，我们使用这些结果来表征功能字段中乘法函数的部分总和的限制行为，尤其是在$ \ mathbb {f} _q [t] $上求解了ERD的差异问题的“校正”形式。
We develop an approach to study character sums, weighted by a multiplicative function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)χ(G)ξ(G), \end{equation} where $χ$ is a Dirichlet character and $ξ$ is a short interval character over $\mathbb{F}_q[t].$ We then deduce versions of the Matomäki-Radziwill theorem and Tao's two-point logarithmic Elliott conjecture over function fields $\mathbb{F}_q[t]$, where $q$ is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the Möbius function for various values of $q$. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that $q$ is a power of $2$. As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a "corrected" form of the Erdős discrepancy problem over $\mathbb{F}_q[t]$.
