关于一些数列的偏差与伪随机性

设p为奇素数.定义xn={{nk+(n)k/p},如果p(|＼)n;0,如果p|n,以及en={+1, 如果p(|＼)n且0≤{nk+(n)k}<1/2;-1, 如果p(|＼)n且1/2≤{nk+(n)k}<1;+1, 如果p|n.其中是n关于模p的乘法逆,满足1≤(n) ≤p-1.利用解析方法研究了数列{xn}和{en}的性质,并证明了{en}是好的伪随机二进制数列.

On the Discrepancy and Pseudorandomness of Some Sequences

Let p be an odd prime number. Define x_n={{(n~k+(n)~k)/p},if p(|＼)n;0,if p|n,and e_n={+1, if p(|＼)n and 0≤{n~k+(n)~k}<1/2;-1, if p(|＼)n and 1/2≤{n~k+(n)~k}<1;+1,if p|n, where (n) is the multiplicative inverse of n modulo p with 1≤(n) ≤p-1. Properties of {x_n} and {e_n} are studied,and it is proved that {e_n} is a good pseudorandom sequence.