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An ErdO¨ s-Kac Type Theorem in Short Intervals Weighted by dk (n)
DOI: 10.12677/PM.2019.99133, PDF, HTML, XML, 下载: 581  浏览: 2,431

Abstract: Let dk (n) be the k-fold divisor function. In this paper, we prove a weighted Erdös-Kac type theo-rem with weight dk (n) in short intervals. This generalizes a recent result of K. Liu and J. Wu.

1. 介绍

${d}_{k}\left(n\right)=\frac{\left({\alpha }_{1}+k-1\right)!}{{\alpha }_{1}!\left(k-1\right)!}\cdots \frac{\left({\alpha }_{s}+k-1\right)!}{{\alpha }_{s}!\left(k-1\right)!}.$ (1)

$k=2$ 时， ${d}_{k}\left(n\right)$ 即为经典除数函数 $d\left(n\right)$

$x\ge 2$ 时，定义 ${d}_{k}\left(n\right)$ 的和函数

${D}_{k}\left(x\right):=\underset{n\le x}{\sum }{d}_{k}\left(n\right)\text{ }\text{ }.$

Landau [2] 和Voronoi [3] 证明了

${D}_{k}\left(x\right)=x{P}_{k-1}\left(x\right)+{\Delta }_{k}\left(x\right)$

${\Delta }_{k}\left(x\right){\ll }_{\epsilon }{x}^{\frac{k-1}{k+1}+\epsilon },$

${\Delta }_{k}\left(x\right)\ll {x}^{\frac{k-1}{k-2}+\epsilon }.$

$\underset{x

$k\ne 4$ 时，目前短区间上相应问题的最佳结果可由 ${D}_{k}\left(x\right)$ 的渐进公式(“长区间”)推得。

${\pi }_{l,k}\left(x,y\right):=\underset{\begin{array}{l}x

${\pi }_{l,k}\left(x,y\right)=\frac{y}{\mathrm{log}x}\frac{{\left(k\mathrm{log}\mathrm{log}x\right)}^{l-1}}{\left(l-1\right)!}\left\{\lambda \left(\frac{l-1}{k\mathrm{log}\mathrm{log}x}\right)+{O}_{B,\epsilon }\left(\frac{\mathrm{log}\mathrm{log}x}{l\mathrm{log}x}+\frac{l-1}{{\left(\mathrm{log}\mathrm{log}x\right)}^{2}}\right)\right\},$

$x\ge 2$${x}^{7/12+\epsilon }\le y\le x$$1\le l\le Bk\mathrm{log}\mathrm{log}x$ 一致成立，这里

$\lambda \left(z\right)=\frac{k}{\Gamma \left(kz+1\right)}\underset{p}{\prod }\left(1+\underset{v\ge 1}{\sum }\frac{\left(v+k-1\right)!}{v!\left(k-1\right)!{p}^{v}}z\right){\left(1-\frac{1}{p}\right)}^{kz},$

1939年，Erdös和Kac [9] 证明了 $\omega \left(n\right)$ 的概率分布，对于每一个 $\lambda \in R$，他们证明了如下的中心极限定理：

$\frac{1}{x}\underset{\begin{array}{c}n\le x\\ \omega \left(n\right)-\mathrm{log}\mathrm{log}x\le \lambda \sqrt{\mathrm{log}\mathrm{log}x}\end{array}}{\sum }1\to \Phi \left(\lambda \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\to \infty$

$\Phi \left(\lambda \right):=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\lambda }{\text{e}}^{-{\tau }^{2}/2}\text{d}\tau .$

2015年，Elliott [10] [11] 证明了如下权为 $d{\left(n\right)}^{\alpha }$ ( $\alpha \in R$ )的中心极限定理，即对每一个 $\lambda \in R$，有

$\frac{1}{{D}_{\alpha }\left(x\right)}\underset{\begin{array}{c}n\le x\\ \omega \left(n\right)-{2}^{\alpha }\mathrm{log}\mathrm{log}x\le \lambda \sqrt{{2}^{\alpha }\mathrm{log}\mathrm{log}x}\end{array}}{\sum }d{\left(n\right)}^{\alpha }\to \Phi \left(\lambda \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\to \infty$

${D}_{\alpha }\left(x\right):=\underset{n\le x}{\sum }d{\left(n\right)}^{\alpha }$

$\frac{1}{D\left(x,y\right)}\underset{\begin{array}{c}x

$D\left(x,y\right):=\underset{x

O-符号中的隐含常数只依赖于 $\epsilon$，并且误差项的上界估计是最优的。

N为全体自然数集，R为全体实数集，C为全体复数集； $\Gamma \left(z\right)$ 伽马函数；Landau符号， $f\left(x\right)=O\left(g\left(x\right)\right)$$f\left(x\right)\ll g\left(x\right)$ 是指存在常数 $c>0$，使得 $|f\left(x\right)|\le cg\left(x\right)$$\underset{p}{\prod }\text{ }$ 表示p遍历所有素数并求乘积； $f\left(x\right)\sim g\left(x\right)$ 是指 $\underset{x\to \infty }{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}=1$

2. 预备知识

$F\left(s\right):=\underset{n=1}{\overset{\infty }{\sum }}f\left(n\right){n}^{-s}.$

$z\in C$$\omega \in C$$\alpha >0$$\delta \ge 0$$A\ge 0$$B>0$$C>0$$M>0$ 为常数， $s=\sigma +i\tau$。如果 $F\left(s\right)$ 满足下列条件，则称其是 $P\left(z,\omega ,\alpha ,\delta ,A,B,C,M\right)$ 型的：

(a) 对于任意的 $\epsilon >0$，有

$|f\left(n\right)|{\ll }_{\epsilon }M{n}^{\epsilon }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n\ge 1\right)$

$\ll$ -中的隐含常数只与 $\epsilon$ 有关。

(b) 当 $\sigma >1$ 时有

(c) Dirichlet级数

O-符号中的隐含常数只依赖于A，B，

，及一致成立，其中

(2)

，当时，函数是可乘函数，其在(p为素数，)的值可由下列公式给出

(3)

3. 定理1.1的证明

(4)

(5)

(6)

(7)

(8)

，可得

4. 定理1.2的证明

，记的特征函数，我们有

(9)

，由引理2.3，计算得到下面结果：

(10)

，我们得到

(11)

(12)

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