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
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.
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