Some new hybrid power mean formulae of trigonometric sums

Li Chen,Zhuoyu Chen

doi:10.1186/s13662-020-02660-7

Abstract

We apply the analytic method and the properties of the classical Gauss sums to study the computational problem of a certain hybrid power mean of the trigonometric sums and to prove several new mean value formulae for them. At the same time, we also obtain a new recurrence formula involving the Gauss sums and two-term exponential sums.

Highlights

For any integer m and odd prime p ≥ 3, the cubic Gauss sums A(m, p) = A(m) are defined as follows: p–1 ma3 A(m) = e, p a=0 where, as usual, e(y) = e2πiy
We found that several scholars studied the hybrid mean value problems of various trigonometric sums and obtained many interesting results
We do not know whether there exists a precise computational formula for (2), where c is any integer with (c, p) = 1, and p ≡ 1 mod 3

Summary

Introduction

1 Introduction For any integer m and odd prime p ≥ 3, the cubic Gauss sums A(m, p) = A(m) are defined as follows: p–1 ma3 We found that several scholars studied the hybrid mean value problems of various trigonometric sums and obtained many interesting results. For p ≡ 1 mod 3, they proved an interesting third-order linear recurrence formula for We do not know whether there exists a precise computational formula for (2), where c is any integer with (c, p) = 1, and p ≡ 1 mod 3.

Objectives

Results

Conclusion