短区间的并集中整数及其m次幂的差的均值分布

王晓瑛, 曹艳梅

数学学报 ›› 2018, Vol. 61 ›› Issue (6) : 943-950.

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数学学报 ›› 2018, Vol. 61 ›› Issue (6) : 943-950. DOI: 10.12386/A2018sxxb0085
论文

短区间的并集中整数及其m次幂的差的均值分布

    王晓瑛, 曹艳梅
作者信息 +

On the Mean Value Distribution of the Difference Between an Integer and Its m-th Power over Unions of Short Intervals

    Xiao Ying WANG, Yan Mei CAO
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摘要

本文研究了短区间的并集中整数及其m次幂的差的均值分布问题,给出了渐近公式.具体来说,设p是奇素数,1 ≤ Hp,实数δ满足0 < δ ≤ 1,整数m ≥ 2.设Ij是(0,p)的互不相交的子区间,1 ≤ jJ,满足H/2 ≤|Ij|≤ H,以及(yp表示y在模p下的非负最小剩余.定义I=Uj=1JIj,并设χ是模p的Dirichlet非主特征.证明了

以及
.

Abstract

We study the mean value distribution of the difference of an integer and its m-th power over unions of short intervals, and give some asymptotic formulas. For details, let p be an odd prime, 1 ≤ Hp, 0 < δ ≤ 1 be any fixed real number, and m ≥ 2 be integers. Let I(j) be disjoint subintervals of (0, p), 1 ≤ jJ, satisfying H/2 ≤ |I(j)| ≤ H, and let (y)p denote the non-negative least residue of y modulo p.Define I=Uj=1J I(j), and let χ be the Dirichlet character modulo p. We prove that


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..

关键词

整数及其m次幂 / 二项指数和 / 短区间 / 特征

Key words

integer and its m-th power / two-term exponential sum / short interval / character

引用本文

导出引用
王晓瑛, 曹艳梅. 短区间的并集中整数及其m次幂的差的均值分布. 数学学报, 2018, 61(6): 943-950 https://doi.org/10.12386/A2018sxxb0085
Xiao Ying WANG, Yan Mei CAO. On the Mean Value Distribution of the Difference Between an Integer and Its m-th Power over Unions of Short Intervals. Acta Mathematica Sinica, Chinese Series, 2018, 61(6): 943-950 https://doi.org/10.12386/A2018sxxb0085

参考文献

[1] Browning T. D., Haynes A., Incomplete Kloosterman sums and multiplicative inverses in short intervals, Int. J. Number Theory, 2013, 9:481-486.
[2] Cochrane T., Zheng Z., Upper bounds on a two-term exponential sum, Sci. China Ser. A, 2001, 44:1003-1015.
[3] Liu H., Zhang Z., A generalization of Woods problem in unions of short interval (in Chinese), Sci. China Ser. A, 2017, 9:467-478.
[4] Xu Z., Distribution of the difference of an integer and its m-th power mod n over incomplete intervals, J. Number Theory, 2013, 133:4200-4223.
[5] Zhang W., On a problem of A. C. Woods, Adv. Math. China, 1994, 23:284-285.
[6] Zhang W., On the difference between an integer and its inverse module n, J. Number Theory, 1995, 52:1-6.
[7] Zhang W., Some estimates of trigonometric sums and their application, Acta Math. Hungar., 1997, 76:17-30.

基金

国家自然科学基金资助项目(11571277);陕西省工业科技攻关项目(2016GY-077)

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