论文标题
$ k $ free有效分隔线的分布和功能字段中的节省函数
The Distribution of $k$-Free Effective Divisors and the Summatory Totient Function in Function Fields
论文作者
论文摘要
我们研究了在经典环境中对$ k $ free指标和基本功能进行研究的动机，我们研究了它们的功能字段类似物。首先，我们根据相关的Zeta函数的零来得出插入函数的误差项的表达式。在线性独立性假设下，我们明确构建了这些误差项的限制分布，并计算出实际$β> 0 $的间隔$ [-β，β] $以它们的频率。我们还表明，这些误差项是公正的，也就是说，它们是正面的，并且经常是否定的。最后，我们检查了固定属的高纤维曲线家族中这些误差术语的平均行为。我们通过遵循Cha和Humphries发起的一般框架来获得这些结果。
Motivated by the study of the summatory $k$-free indicator and totient functions in the classical setting, we investigate their function field analogues. First, we derive an expression for the error terms of the summatory functions in terms of the zeros of the associated zeta function. Under the Linear Independence hypothesis, we explicitly construct the limiting distributions of these error terms and compute the frequency with which they occur in an interval $[-β, β]$ for a real $β> 0$. We also show that these error terms are unbiased, that is, they are positive and negative equally often. Finally, we examine the average behavior of these error terms across families of hyperelliptic curves of fixed genus. We obtain these results by following a general framework initiated by Cha and Humphries.
