| 李英杰.序列[n~c]上多维除数函数的和[J].数学年刊A辑,2011,32(3):355~364 |
| 序列[n~c]上多维除数函数的和 |
| The Sum of Multidimensional Divisor Function on a Sequence of Type[n~c] |
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| 中文关键词: 除数函数 渐近公式 指数和 指数对 |
| 英文关键词:Divisor function Asymptotic formula Exponential sum Exponent pair |
| 基金项目:上海高校选拔培养优秀青年教师科研专项基金(No.ssc08017); 上海海洋大学博士科研启动基金资助的项目 |
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| 中文摘要: |
| 设$[\theta]$表示$\theta$的整数部分, $k \ge 2$, $d_k (n)$为除数函数. 证明了当实数$c$满足$1 < c < \frac{3849}{3334}$时,
$\sum\limits_{n \le x}d_k([n^c])$具有渐近公式, 从而改进了吕广世和翟文广的结果($1 < c < \frac{495}{433}$),
而且当$k=2$时, 实数$c$的范围可以改进到$1 < c < \frac{391}{335}$. |
| 英文摘要: |
| Let $[\theta]$ be the integral part of $\theta$ and $k \ge 2$, $d_k(n)$ denote the divisor function. In this paper it is proved that
$\sum\limits_{n \le x}d_k([n^c])$ has an asymptotic formula when $1 < c < \frac{3849}{3334}$, which improves L\"u Guangshi and
Zhai Wenguang's result $1 < c < \frac{495}{433}$. Moreover, if $k=2$, then the range of $c$ can be enlarged to $1 < c < \frac{391}{335}$. |
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