An analytic research on "estimates of character sums" and "the distribution property of primitive roots"

本文研究了“特征标和的估计”和“原根的分布性质”。在特征标和的估计方面，我们考虑了特征标和S(X；0，N)的一些平均，其中S(X；0，N)=Σ^N_&lt；n=0&gt；X(N)，得到了|S(X；0，N)|的平均值的一个新的上界。我们的界是著名的Polya-Vinogradov界和Burgess界在平均意义上的改进。作为应用，我们得到了关于L(1/2+it，X)的q-估计的一些平均型结果.设a是[双正]1的正整数，q_a(x；t，S)是素数p[小于或等于]x的集合，使得a(Modp)在群(Z/pz)^*中的剩余阶与S模t同余.我们知道，剩余序的波动是非常不规则的，到目前为止我们对剩余序的分布性质知之甚少.在这项研究中，我们计算了当i=0，1，2，3时Q_a(x；4，i)的自然密度(与K.Chinen博士合作)。我们的主要结果表明，例如，当a是无平方分布且≡为1l(Mod4)时，上述分布具有一个美丽的性质：无条件地得到Q_a(x；4，0)和Q_a(x；4，2)=1/3的自然密度，在广义黎曼假设下得到Q_a(x；4，1)和Q_a(x；4，3)=1/6的自然密度。

In this research, we studied about "estimates of character sums" and "the distribution property of primitive roots".On estimates of character sums, we considered some averages of the character sum S(X ; 0, N), where S(X ; 0, N)=Σ^N_<n=0>X(n), and we got a new upper bound for the average value of |S(X ; 0, N)|. Our bound is an improvement of the famous Polya-Vinogradov's bound and Burgess' bound, in the sense of average. As an application of this new bound, we obtained some average type results on the q-estimate of L(1/2+it,X).Let a be a positive integer with a【double plus】1 and Q_a(x ; t,s) be the set of primes p【less than or equal】x such that the residual order of a(mod p) in the group (Z/pZ)^* is congruent to s modulo t. It is known that the residual order fluctuates quite irregularly and we know only little about the distribution property of the residual order so far. In this research we calculated the natural densities of Q_a(x ; 4, i) for i=0, 1, 2, 3 (Collaboration with Dr. K. Chinen). Our main result shows that, for example, when a is square-free and ≡1l(mod 4), then the above distribution has a beautiful property:The natural density of Q_a(x ; 4,0) and Q_a(x ; 4,2) =1/3, unconditional result,The natural density of Q_a(x ; 4, 1) and Q_a(x ; 4,3) =1/6, under Generalized RiemannHypothesis.We got similar results fore more general a's.