Upper bounds on character sums with rational function entries, preprint

Abstract. We obtain formulae and estimates for character sums of the type S(χ, f, pm) = ∑pm x=1 χ(f(x)), where p m is a prime power with m ≥ 2, χ is a multiplicative character (mod pm), and f = f1/f2 is a rational func-tion over Z. In particular, if p is odd, d = deg(f1) + deg(f2) and d ∗ = max(deg(f1), deg(f2)) then we obtain |S(χ, f, pm) | ≤ (d − 1)pm(1− 1 d ∗ ) for any non constant f (mod p) and primitive character χ. For p = 2 an extra facto