Upper Bounds on Character Sums with Rational Function Entries

Abstract 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 ≥ 2, χ is a multiplicative character (mod pm), and f=f1/f2 is a rational function over ℤ. In particular, if p is odd, d=deg(f1)+deg(f2) and d* = max(deg(f1), deg(f2)) then we obtain $$ {\left| {S{\left( {\chi ,f,p^{m} } \right)}} \right|} \leqslant {\left( {d - 1} \right)}p^{{m{\left( {1 - \frac{1} {{d*}}} \right)}}} $$ for any non-constant f (mod p) and primitive character χ. For p = 2 an extra factor of $$ 2{\sqrt 2 } $$ is needed.