Analytic study on distribution properties of the residual order and residual index, and on the relation between the distribution and estimates on character sums

设a为自然数，p为素数，D_a(p)和I_a(p)分别表示类a（mod p）的残差阶数和残差指标。D_a(p)和I_a(p)都是从p（所有素数的集合）到N（所有自然数的集合）的满射映射。本文研究了映射D_a(p)的分布性质。首先考虑集合：Q_a(x; s, t){p≦x；D_a(p)≡s (mod t)}， s和t为自然数，得到结果；1)当我们假设广义黎曼假设时，我们可以证明集合Q_a（x; s,t）的自然密度Δ_a（s,t）的存在性；2)我们可以有效地计算密度Δ_a(s,t)，并且通过一些数值和理论观察，我们发现数论函数Δ_a（s,t）的一些有趣的性质。其次，我们考虑集合：M(x) = {p≦x；D_2(p)是质数。然后，在g.r.h.下，我们有一个估计#M(x) << x(log x)^（-2）。这一估计很可能是“最佳可能”。

Let a be a natural number, p be a prime number, D_a(p) and I_a(p) denote the residual order and the residual index of the class a(mod p), respectively.Both D_a(p) and I_a(p) are surjective map from P(the set of all primes) to N(the set of all natural numbers). We studied here the distribution properties of the map D_a(p).Firstly, we considered the set : Q_a(x ; s, t){p≦x ; D_a(p)≡s (mod t)}, s and t be natural numbers, and we obtained the results ;1)when we assume the generalized Riemann Hypothesis, we can prove the existence of the natural density Δ_a(s,t) of the set Q_a(x ; s, t),2)we can calculate the density Δ_a(s, t) effectively, and through some numerical and theoretical observations, we found out some interesting properties of the number theoretical function Δ_a(s, t).Secondly, we considered the set : M(x) = {p ≦ x ; D_2(p) is a prime}. Then, under G.R.H., we have an estimate #M(x) << x(log x)^(-2). This estimate is most likely to be "best possible".