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

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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.