短区间的并集中整数及其m次幂的差的均值分布
On the Mean Value Distribution of the Difference Between an Integer and Its m-th Power over Unions of Short Intervals
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 ≤ H ≤ p,实数δ满足0 < δ ≤ 1,整数m ≥ 2.设I(j)是(0,p)的互不相交的子区间,1 ≤ j ≤ J,满足H/2 ≤|I(j)|≤ H,以及(y)p表示y在模p下的非负最小剩余.定义I=Uj=1JI(j),并设χ是模p的Dirichlet非主特征.证明了
以及
.
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 ≤ H ≤ p, 0 < δ ≤ 1 be any fixed real number, and m ≥ 2 be integers. Let I(j) be disjoint subintervals of (0, p), 1 ≤ j ≤ J, 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
, and
..
整数及其m次幂 / 二项指数和 / 短区间 / 特征 {{custom_keyword}} /
integer and its m-th power / two-term exponential sum / short interval / character {{custom_keyword}} /
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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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