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Upper Bounds on Character Sums with Rational Function Entries

Authors:	Todd Cochrane Chun Lei Liu Zhi Yong Zheng

Institution:	(1) Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA;(2) Department of Mathematics, Zhengzhou University, Zhengzhou 450002, P. R. China;(3) Department of Mathematics, Tsinghua University, Beijing 100084, P. R. China

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 p ^m is a prime power with m ≥ 2, χ is a multiplicative character (mod p ^m), and f=f ₁/f ₂ is a rational function over ℤ. In particular, if p is odd, d=deg(f ₁)+deg(f ₂) and d* = max(deg(f ₁), deg(f ₂)) 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. The second and third authors are supported by the National Science Foundation of the P. R. C., and the third author is partially supported by the 973 Project.

Keywords:	Exponential sums Character sums
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