High order moments of character sums
High order moments of character sums
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DOI：
10.1090/s0002-9939-98-04625-5
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发表时间：
1998
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通讯作者：
Todd Cochrane;Zhiyong Zheng
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文献类型：
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作者：
Todd Cochrane;Zhiyong Zheng
我们建立了上界a+B，1Z>；Xx(X)12？EKP L+E+BkpE，p_Ei Ex()L<；&Lt；e，kp-e+gX？xOx=a+L，其中p是素数，k是任意正整数，其和覆盖所有非主乘特征标(Mod P)。1.本文得到了特征标和a+B(1)_1 SE SE X(X)|X#XOx=a+L的上界，其中a，B，k是正整数，p是素数，X贯穿于可乘特征标集合(Modp)，xi是主特征标。我们将假设B<；p，且区间a+1<；x<；A+B不包含p的倍数。由Polya-Vinogradov不等式直接推出的(1)中和的一个平凡界是a+B1 S E(X)12k<；Pk(Logp)2k。X#XO x=a+1设B为立方(2)B={xEZ2k：A+1<；xi<；A+B，1<；i<；2k}，V是同余X1X2..XkXk+1Xk+2的整数解集.X2k(Mod P)。然后A+B a+B IBnVI 1 Z X(XlX2...-XkX4i.Xj)|8nv|=1 E E E%xx。Kk+L，X2k)XL=a+L X2k=a+L X编辑：1996年2月25日。1991年数学科目分类。主11L40、11D79。
We establish the upper bound a+B ,1 Z > X x(X)12 ?ekP l+E+BkpE, p_ EI Ex()l<<e ,k p-e+ g X?Xo x=a+l with p a prime and k any positive integer, the sum being over all nonprincipal multiplicative characters (mod p). 1. In this paper we obtain upper bounds on the character sum a+B (1) _ 1 SE SE X(X)| X#Xo x=a+l where a, B and k are positive integers, p is a prime, X runs through the set of multiplicative characters (mod p), and Xi is the principal character. We shall assume that B < p and that the interval a + 1 < x < a + B does not contain a multiple of p. A trivial bound for the sum in (1) that follows directly from the Polya-Vinogradov inequality is a+B 1 S E (X)12k < Pk(log p)2k. X#Xo x=a+1 Let B be the cube (2) B = {x EZ2k: a + 1 < xi < a + B, 1 < i < 2k} of cardinality BI = B2k, and let V be the set of integer solutions of the congruence X1X2 ..Xk Xk+1Xk+2 ... X2k (mod p). Then a+B a+B IBnVI 1 Z X(XlX2...-XkX4i. xj ) |8nv|= 1 E E E%xx . kk+l, X2k ) xl=a+l X2k=a+l X Received by the editors February 25, 1996. 1991 Mathematics Subject Classification. Primary 11L40, 11D79.