|
| 摘要: |
| 本文利用不完全 Kloosterman 和的均值定理来研究了短区间的并集中的 D. H. Lehmer 问题, 并且给出了渐近公式. 设 $p$ 是奇素数, $H>0$, $K>0$, 并设
$I_1^{(j)}$, $I_2^{(j)}$ 是 $(0,p)$ 的子区间, $1\leq j\leq J$, 满足 $|I_1^{(j)}|=H$, $|I_2^{(j)}|=K$, 以及 $I_1^{(j)}\bigcap I_1^{(k)}=\emptyset$, 当 $j\neq k$ 时. 设 $c$, $n$ 为整数, 满足 $n\geq 2$ 以及 $(n,p)=(c,p)=1$. 本文证明了
\begin{eqnarray*}
\sum_{j=1}^{J}\mathop{\mathop{{\mathop{{\sum}}_{x\in\I_1^{(j)}}\mathop{{\sum}}_{y\in\I_2^{(j)}}}}_{xy\equiv
c (\bmod p)}}_{n\nmid(x+y)}1 =\left(1-\frac{1}{n}\right)\frac{JHK}{p}+O\left(J^{\frac{1}{2}}p^{\frac{1}{2}}\log p\log H\right).
\end{eqnarray*} |
| 关键词: D. H. Lehmer 问题 不完全 Kloosterman 和 短区间 |
| DOI： |
| 分类号:O156.4. |
| 基金项目:国家自然科学基金项目（面上项目，重点项目，重大项目） |
|
| On a generalization of the D. H. Lemher problem in unions of short intervals |
|
WANG Xiao-ying
|
| Abstract: |
| In this paper we study the D. H. Lemher problem in unions of short intervals by using the mean value theorem for incomplete Kloosterman
sums, and give an asymptotic formula. For details, let $p$ be an odd prime, $H>0$, $K>0$, and let $I_1^{(j)}$, $I_2^{(j)}$ be subintervals of $(0,p)$, $1\leq j\leq J$, satisfying $|I_1^{(j)}|=H$, $|I_2^{(j)}|=K$, and $I_1^{(j)}\bigcap I_1^{(k)}=\emptyset$ for $j\neq k$. Assume that $c$, $n$ are
integers with $n\geq 2$ and $(n,p)=(c,p)=1$. We prove that
\begin{eqnarray*}
\sum_{j=1}^{J}\mathop{\mathop{{\mathop{{\sum}}_{x\in\I_1^{(j)}}\mathop{{\sum}}_{y\in\I_2^{(j)}}}}_{xy\equiv
c (\bmod p)}}_{n\nmid(x+y)}1 =\left(1-\frac{1}{n}\right)\frac{JHK}{p}+O\left(J^{\frac{1}{2}}p^{\frac{1}{2}}\log p\log H\right).
\end{eqnarray*} |
| Key words: D. H. Lemher problem incomplete Kloosterman sum short interval |
