摘要
We obtain formulae and estimates for character sums of the type $S\left( {\chi ,f,p^m } \right) = \sum\nolimits_{x = 1}^{p^m } {\chi \left( {f\left( x \right)} \right),} $ where pm is a prime power with m S 2, L is a multiplicative character (mod p^m), and f=f1/f2 is a rational function over ê. In particular, if p is odd, d=deg(f1)+deg(f2) and d* = max(deg(f1), deg(f2)) then we obtain $\left| {S\left( {\chi ,f,p^m } \right)} \right| \le \left( {d - 1} \right)p^{m\left( {1 - {1 \over {d*}}} \right)}$ for any non-constant f (mod p) and primitive character L. For p = 2 an extra factor of $2\sqrt 2$ is needed.
We obtain formulae and estimates for character sums of the type $S\left( {\chi ,f,p^m } \right) = \sum\nolimits_{x = 1}^{p^m } {\chi \left( {f\left( x \right)} \right),} $ where pm is a prime power with m S 2, L is a multiplicative character (mod p^m), and f=f1/f2 is a rational function over ê. In particular, if p is odd, d=deg(f1)+deg(f2) and d* = max(deg(f1), deg(f2)) then we obtain $\left| {S\left( {\chi ,f,p^m } \right)} \right| \le \left( {d - 1} \right)p^{m\left( {1 - {1 \over {d*}}} \right)}$ for any non-constant f (mod p) and primitive character L. For p = 2 an extra factor of $2\sqrt 2$ is needed.
