关于D.H.Lehmer问题的一个推广

A generalizationon a problem of D.H.Lehmer

对任意与q互素且满足1≤a≤q的整数a,存在唯一一个整数b满足1≤b≤q使得ab≡1modq.D.H.Lehmer建议研究2不整除a-b时的情形,本文中用任意大于2的正整数γ来代替2,并利用Kloosterman和的估计以及三角和的性质,给出了一个有趣的渐近公式.

For each integer a with 1≤a≤q and (a,q)=1,there exists one and only one b with 1≤b≤q such that ab≡1 mod q.Professor D.H.Lehmer proposed the problem of investigating the number of cases in which 2a-b.2 is replaced by any positive integer γ≥2,and an interesting asymptotic formula is given through using the estimates of Kloosterman sums and the properties of trigonometric sums.