关于整数幂模q剩余之差的高次均值问题

发布时间：2019-05-13 18:18

【摘要】：整数及其逆的分布问题一直受到众多数论学者的广泛关注,人们对此进行了深入研究,得到了丰富的成果.本文引入了不同整数幂模q剩余之差的分布问题,这是对整数及其逆的分布问题的一种推广.文章主要利用初等数论、解析数论中的一些经典的方法,并结合三角和的性质及两项指数和的估计,研究了整数幂模q剩余之差的高次均值问题,得到了好的渐近公式.具体研究结果如下:令p为奇素数,α为正整数,q= pα,m1,m2为不相等的正整数常数,0δ,λ1,λ2≤1为实数,k为非负的任意整数.1.令a为满足1 ≤ a ≤ q,(a,q)= 1的整数,则存在唯一的整数b满足1≤b ≤ g及其b三am1(mod q),记为(am1)q,即整数am1模q的最小正剩余.当q[1/δ]时,研究了整数幂模q剩余之差的高次均值分布,得到了渐近公式其中Σ'表示与q互素的整数之和,ε0为任意实数,O所包含的常数与δ,m1,m2,k有关;2.当qmax{[1/λ1],[1/λ2}时,研究了在不完整区间上Lehmer问题的推广,渐近公式如下.其中O所包含的常数与λ1,λ2,m1,m2,κ有关.
[Abstract]:The distribution of integers and their inverses has been widely concerned by many number theorists, which has been deeply studied and rich results have been obtained. In this paper, the distribution problem of the difference between different integer power modules Q is introduced, which is a generalization of the distribution problem of integers and their inverses. In this paper, by using the elementary number theory and some classical methods in the number theory, combined with the properties of the triangular sum and the estimation of the two exponential sums, the higher order mean value problem of the difference between the integers power module Q residue is studied, and a good asymptotic formula is obtained. The concrete results are as follows: let p be odd prime, 伪 be positive integer, Q = p 伪, M1, m2 be unequal positive integer constant, 0 未, 位 1, 位 2 鈮，

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