短区间的并集中整数及其m次幂的差的均值分布

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=∪_(j=1)~JI~((j)),并设X是模p的Dirichlet非主特征.证明了Σ x∈1 |x-(x~m)p|δp 1=1/p∫_0~(δp]) (Σ x∈1 x≤p-1-t 1+Σ x∈1 x≥t=1 1)dt+O(mJ~(1/2)P~(1/2)log~2 plog H),以及Σ x∈1 |x-(x~m)p|δp X(x)mJ~(1/2)P~(1/2)log~2 plog H.

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

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=U_j=1^J I^(j), and let χ be the Dirichlet character modulo p. We prove that , and ..