一个特殊的Gauss和以及它的上界估计

付瑞琴, 杨海

数学学报 ›› 2014, Vol. 57 ›› Issue (6) : 1141-1146.

PDF(359 KB)
PDF(359 KB)
数学学报 ›› 2014, Vol. 57 ›› Issue (6) : 1141-1146. DOI: 10.12386/A2014sxxb0104
论文

一个特殊的Gauss和以及它的上界估计

    付瑞琴1,2, 杨海3,4
作者信息 +

A Special Gauss Sums and Its Upper Bound Estimate

    Rui Qin FU1,2, Hai YANG3,4
Author information +
文章历史 +

摘要

p 是一个奇素数.对任意满足1≤ap-1的整数 a, 存在唯一的整数1≤ap-1, 使得 a ·a≡1mod p成立.设 N(p)表示区间 1≤ap-1中所有满足条件a与a具有相反奇偶性的a的集合.本文利用解析方法以及广义 Kloosterman和∑的性质研究一类特殊的Gauss和∑aN(p) χ(a)e((ma)/p)的估计问题, 给出一个较强的上界估计,其中e(x)=e2π#/emem#x,(m,p)=1, 且 χ是模p的任意特征.

Abstract

Let p be an odd prime.For each integer a with 1≤ap-1, we know that there exists one and only one a with 1≤ap-1 such that a ·a ≡1 mod p.Let N(p) denote the set of all integers 1 ≤ a ≤ p -1 in which a and a are of opposite parity.The main purpose of this paper is using the analytic method and the properties of general Kloosterman sums to study the estimate problem of the Gauss sums ∑aN(p) χ(a)e((ma)/p), and give a sharp upper bound estimate for it, where e(x)=e2π#/emem#x,(m,p)=1, and χ denotes any character mod p.

关键词

Lehmer数 / Gauss和 / 广义Kloosterman和 / 上界估计

Key words

Lehmer numbers / Gauss sums / general Kloosterman sums / upper bound estimate

引用本文

导出引用
付瑞琴, 杨海. 一个特殊的Gauss和以及它的上界估计. 数学学报, 2014, 57(6): 1141-1146 https://doi.org/10.12386/A2014sxxb0104
Rui Qin FU, Hai YANG. A Special Gauss Sums and Its Upper Bound Estimate. Acta Mathematica Sinica, Chinese Series, 2014, 57(6): 1141-1146 https://doi.org/10.12386/A2014sxxb0104

参考文献

[1] Apostol T.M., Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.

[2] Conrad K., On Weil's proof of the bound for Kloosterman sums, J.Number Theory, 2002, 130(2): 439-446.

[3] Funakura T., On Kronecker's limit formula for Dirichlet series with periodic coefficients, Acta Arithmetica, 1990, 55(1): 59-73.

[4] Guy R.K., Unsolved Problems in Number Theory (Second Edition), Springer-Verlag, New York, 1994.

[5] Xi P., Yi Y., On character sums over flat numners, J.Number Theory, 2010, 130(5): 1234-1240.

[6] Zhang W.P., On a problem of D.H.Lehmer and its generalization, Composito Math., 1993, 86(3): 307-316.

[7] Zhang W.P., A problem of D.H.Lehmer and its generalization (II), Compositio Math., 1994, 91(1): 47-56.

[8] Zhang W.P., A problem of D.H.Lehmer and its mean square value formlua, Japanese J.Math., 2003. 29(1): 109-116.

[9] Zhang W.P., A mean value related to D.H.Lehemr's problem and the Ramanujan's sum, Glasgow Math. J., 2012, 54(1): 155-162.

基金

国家自然科学基金项目(11226038, 11371012);陕西省教育厅专项基金(11Jk0472,11Jk0474);西安工程大学博士科研基金(BS1016)

PDF(359 KB)

Accesses

Citation

Detail

段落导航
相关文章

/