Pure and mixed exponential sums

Abstract

We obtain explicit formulae and sharp estimates for pure and mixed exponential sums of the type S(f, pm) = ∑pm x=1 epm (f(x)) and S(χ, f, pm) = ∑pm x=1 χ(x)epm (f(x)), where p m is a prime power with m ≥ 2, χ is a multiplicative character (mod pm), and f is a polynomial. For nonconstant f (mod p) we prove |S(χ, f, pm)| ≤ 2d 1 M+1 p m(1− 1 M+1 ) where d is the degree of f and M is the maximum multiplicity of the set of critical points associated with the sum. In particular, |S(χ, f, pm)| ≤ 4dp 1 d+1 ) . The exponent in the latter upper bound is shown to be best possible. This solves the problem of generalizing the classical upper bound of Hua for pure exponential sums to mixed exponential sums. When all of the critical points are of multiplicity one then we obtain an explicit formula for the exponential sum, generalizing the classical formula of Salie for the Kloosterman sum.

Locations

• Acta Arithmetica - View - PDF