A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

The main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.


Introduction
Let q ≥ 3 be a fixed integer. For any integers k and m with k ≥ 2 and (m, q) = 1, the kth Gauss sums G(m, k; q) and the two-term exponential sums H(m, k; q) in [1]  where e(y) = e 2π iy and i 2 = -1.
We all know that these sums occupy a very important position in the study of analytic number theory, and many number theory problems are related to them. Therefore, many scholars have studied their various properties and obtained a series of meaningful results.
Recently, Zhang Wenpeng and Chen Zhuoyu [1] studied the hybrid power mean involving H(m, 3; p) and the reciprocal of the quartic Gauss sums G(m, 4; p), and they obtained two interesting results as follows.
If p is a prime with p ≡ 5(mod8), then one has the identity If p is a prime with p ≡ 5(mod8) for any real number k ≥ 0, then one has where α = α(p) = p-1 2 a=1 ( a+a p ) is an integer, ( * p ) denotes the Legendre symbol modp, and a denotes the solution of the equation ax ≡ 1(modp).
These results are significant, because dealing with the reciprocal of the trigonometric sums is not common to us. But the methods in their article cannot handle the case of p ≡ 1(mod8), thus leaving it as an open problem.
Of course, the integer α = α(p) in (1) and (2) is closely related to p. In fact, if p ≡ 1(mod 4), then we have (see ) , and r is any quadratic non-residue modulo p. In this paper, we consider a generalized problem: For any prime p with p ≡ 1(mod8) and number-theoretic function F(m), whether there is an exact calculating formula for the hybrid power mean where k ≥ 0 is an integer. We use the analytic methods and the properties of the fourth character modulo p to give an interesting fourth-order linear recursive formula for V k (p). Theorem 1 Let p be a prime with p ≡ 1(mod8). Then, for any number-theoretic function F(m), we have the fourth-order linear recursive formula
Obviously, in order to obtain all values of V j (p) for any integer k ≥ 0, we need to compute V 0 (p), V 1 (p), V 2 (p), and V 3 (p), then we can compute all the values of V k (p) using this fourth-order linear recursion formula. In general, the first four terms of V j (p) do not always get the exact value, but for some special function F(m), we can compute the exact value of V j (p) with j = 0, 1, 2, 3, and we can get all the terms of the recursive sequence V j (p).
Especially for F(m) = 1 and W k (p) = p-1 m=1 1 G k (m,4;p) in Theorem 1, we have the following result.
Taking k = 2 or 4, from these theorems we have the following corollaries.

Corollary 1
If p is a prime with p ≡ 1(mod8), then we have the identity Corollary 2 If p is a prime with p ≡ 17(mod24), then we have the identity

Corollary 3
If p is a prime with p ≡ 1(mod24), then we have the identity

Several lemmas
To complete the proofs of our theorems, we need to give some basic lemmas. Of course, the proofs of these lemmas need some knowledge of elementary and analytic number theory. They can be found in many number theory books, such as [15][16][17][18]. First we have the following.

Lemma 1
Let p be an odd prime with p ≡ 1( mod 4), ψ be any fourth-order character mod p. Then we have the identity Proof This is Lemma 2.2 in [2]. where ( * p ) denotes the Legendre symbol, and ψ is any fourth-order character modp.
This proves Theorem 1. Note that W 0 (p) = p -1, so Theorem 2 follows from Theorem 1 and Lemma 4. Theorem 3 follows from Lemma 5, Lemma 6, and Lemma 7. Theorem 4 follows from Lemma 5, Lemma 6, and Lemma 8. This completes the proofs of our all results.

Conclusion
The main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problems of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give a series of fourth-order linear recursive formulae. These results not only give the exact values of some special Gauss sums, but they are also some new contribution to the research in related fields.