Upper bounds on a two-term exponential sum*

Abstract

We obtain upper bounds for mixed exponential sums of the type S ({ χ,f,p m )= ∑ pm x=1 X ( x ) e p m ( ax n + bx ) where p m is a prime power with m ≥2 and χ is a multiplicative character (mod p m ). If χ is primitive or p |\ ( a,b ) then we obtain | S ({ χ,f,p m )|≤ If χ is of conductor p and p |\ ( a,b ) then we get the stronger bound | S ({ χ,f,p m )|≤ np m/2