Mean value of mixed exponential sums

On the Hybrid Mean Value of Cochrane Sums and Two-term Exponential Sums

Pure and Mixed Exponential Sums

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 mu...

A Hybrid Mean Value Involving Dedekind Sums and the General Exponential Sums

The main purpose of this paper is using the analytic method, A. Weil's classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums and give a sharp asymptotic formula for it.

A hybrid mean value involving a new Gauss sums and Dedekind sums

‎In this paper‎, ‎we introduce a new sum‎ ‎analogous to Gauss sum‎, ‎then we use the properties of the‎ ‎classical Gauss sums and analytic method to study the hybrid mean‎ ‎value problem involving this new sums and Dedekind sums‎, ‎and‎ ‎give an interesting identity for it.

Mean square value of exponential sums related to representation of integers as sum of two squares

1. Introduction. The problem we address here arises in the study of the error function in the shifted circle problem (see [BCDL]). Let α = (α 1 , α 2) ∈ R