关于Lehmer D H问题的一般化(英文)

On Generalization of Lehmer D H Problem

设p为奇素数。对任意固定正整数k｜p-1及l｜p-1,设A(k)及A(l)分别表示在区间[1,p-1]中k次剩余之集及l次剩余之集,N(k,l,p)表示所有满足a∈A(k),b∈A(l),a和b具有相反的奇偶性且ab≡1(modp)的(a,b)的个数。本文的主要目的是利用三角和估计及广义Kloostermann和估计研究了N(k,l,p)的渐近性质,并得到一个较精确的渐近公式。

Let p be an odd prime. For any fixed positive integers k and l with k|p-1 and l|p-1, let A(k) and A(l) denote the set of all kth residues and lth residues modulo p in interval  respectively, N(k,l,p) denotes the number of all (a,b) with a∈A(k),b∈A(l) for which a and b are of opposite parity and ab≡1(mod p). By using the estimate for trigonometric sums and the esitimate for general Kloostermann sums, we study the asymptotic properties of N(k,l,p), and give a sharper asymptotic formula