Upper Bounds on Character Sums with Rational Function Entries

摘要

We obtain formulae and estimates for character sums of the type S(χ, f P~m)=Σ_(x=1)~(p~m) χ(f(x)), 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 |S(χ, f,p~m)| ≤ (d―1)p~(m(1―1/d~* )) for any non-constant f (mod p) and primitive character χ. For p = 2 an extra factor of 2(2)~(1/2) is needed.