A new fourth power mean of two-term exponential sums

Abstract The main purpose of this paper is to use analytic methods and properties of quartic Gauss sums to study a special fourth power mean of a two-term exponential sums modp, with p an odd prime, and prove interesting new identities. As an application of our results, we also obtain a sharp asymptotic formula for the fourth power mean.


Introduction
Let q ≥ be an integer. For any integer m and n, the two-term exponential sum G(k, h, m, n; q) is de ned as where p is an odd prime, χ is any Dirichlet character mod p and (m, n, p) = . Zhang Han and Zhang Wenpeng [3] proved the identity where p be an odd prime.
Zhang Han and Zhang Wenpeng [4] also obtained where * p = χ denotes the Legendre symbol mod p. Some other related mean value papers can also be found in [5] - [13]. If someone is interested in this eld, please refer to these references. However, regarding the fourth power mean it seems that it hasn't been studied yet, at least so far we haven't seen any related papers. We think one of the reasons for this may be that the methods used in the past are not suitable for studying this situation, or perhaps is not a prime number, so it is di cult to study (1). In this paper we will use analytic methods and properties of quartic Gauss sums to study this problem and solve it completely. That is, we will prove the following two results. Theorem 1. Let p > be a prime with p ≡ mod , then we have Theorem 2. Let p be a prime with p ≡ mod , then we have is an integer satisfying the identity (state displayed identity ), where r is any quadratic non-residue modp : see Theorem 4-11 in [16].
and r is any quadratic non-residue mod p.
From these two theorems we may immediately deduce the following: Corollary. For any odd prime p, we have the asymptotic formula

Several Lemmas
To prove our theorems, we rst need to give several necessary lemmas. Hereafter, we will use many properties of the classical Gauss sums, the fourth-order character mod p and the quartic Gauss sums. All of these contents can be found in any Elementary Number Theory or Analytic Number Theory book, such as references [1], [14] or [16]. These contents will not be repeated here. First we have the followings: Lemma 1. If p is a prime with p ≡ mod , and λ is any fourth-order character mod p, then we have Proof. In fact this is Lemma 2 of [15], so its proof is omitted. Lemma 2. If p is a prime with p ≡ mod , then we have the identity where χ = * p denotes the Legendre's symbol mod p. Proof. First applying trigonometric identity we have (4) Noting the identity λχ = λ and Applying (5) It is clear that the congruences a + b ≡ mod p and a + b ≡ mod p imply that ab a + ab + b ≡ mod p and a + b ≡ mod p. So we have If p ≡ mod , then noting that λ(− ) = we have Applying (5), Lemma 1 and note that τ(λ)τ λ = p we have Combining (3), (4), (6)-(9) we have the identity This proves Lemma 2. (11) From (10), (11) Proof. From identity (2) It is clear that the congruences a + b ≡ c + mod p and a + b ≡ c + mod p imply that (a − ) Now Theorem 2 follows from (18) -(21). Similarly, from Lemma 3, Lemma 5 and the methods of proving Theorem 2 we can also deduce Theorem 1. This completes the proofs of all of our results.