关于非完整的特征和

On Incomplete Character Sums

本文通过研究Ｄｉｒｉｃｈｌｅｔ特征的非完整和来理解其性状．设Ｘ是模ｑ的非平凡的Ｄｉｒｉｃｈｌｅｔ特征，Ｐ为素数，Ｂｕｒｇｅｓｓ证明了ｑ＝ｐａ（ａ＝１，２，４，５或８）时，成立．本文证明了上式在下述条件下亦成立，

Suppose X be a nontrivial Dirichlet character modulo q. Consider the inequality Σ(n=N+1)(N+H) X(n) ≤∈ H1- q+∈. Burgess proved that (i) the inequality holds true if either r≤3or q is cubefree, and (ii) it also holds true when r = 4 and q = pa, a prime power with a = 1, 2, 4, 5 or 8. We are to show through a comprehensive study of the incomplete sums of character that, instead of (ii), the inequality also holds true when r = 4 and q is a positive integer such that for every prime p, ordp(q) = 0, 1, 2, 4, 5 or 8.