指数除数函数在短区间上的Erdős-Kac型定理

Erdős-Kac Type Theorem for Exponential Divisor Function in Short Intervals

摘要: 设$n>1$ 为整数，且$n={\displaystyle \prod _{i=1}^t{p}_i^{{\alpha }_i}}$ ，令${\tau }_k^{\left(e\right)}\left(n\right)={\displaystyle \prod _{p_i^{{\alpha }_i}\left|\right|n}{d}_k\left({\alpha }_i\right)}$ 为指数除数函数，其中$d_k\left(n\right)$ 为$k$ 重除数函数。本文给出了短区间上权为${\tau }_k^{\left(e\right)}\left(n\right)$ 的Erdős-Kac型定理，并得到短区间上${\tau }_k^{\left(e\right)}\left(n\right)$ 均值估计的渐近公式。

Abstract: Let $n>1$ be an integer, $n={\displaystyle \prod _{i=1}^t{p}_i^{{\alpha }_i}}$ , and ${\tau }_k^{\left(e\right)}\left(n\right)={\displaystyle \prod _{p_i^{{\alpha }_i}\left|\right|n}{d}_k\left({\alpha }_i\right)}$ be the exponential divisor function, where $d_k\left(n\right)$ is the k-fold divisor function. In this paper, we establish an Erdős-Kac type theorem with weight ${\tau }_k^{\left(e\right)}\left(n\right)$ in short intervals, and we get an asymptotic formula for the average behavior of ${\tau }_k^{\left(e\right)}\left(n\right)$ in short intervals.