A Hybrid Mean Value Involving the Two-Term Exponential Sums and Two-Term Character Sums

Abstract

The main purpose of this paper is using the properties of Gauss sums and the estimate for character sums to study the hybrid mean value problem involving the two-term exponential sums and two-term character sums and give an interesting asymptotic formula for it.

1. Introduction

Let be an integer and denotes a Dirichlet character . For any integers and with , we define the two-term exponential sum and two-term character sum as follows: where , denotes a nonprincipal Dirichlet character , and is a fixed positive integer.

These sums play a very important role in the study of analytic number theory, so they caused many number theorists’ interest and favor. Some works related to can be found in [1–5]. For example, Cochrane and Zheng [1] show that where denotes the number of all distinct prime divisors of .

On the other hand, the sums are a special case of the general character sums of the polynomials where and are any positive integers and is a polynomial. If is an odd prime, then Weil (see [6]) obtained the following important conclusion.

Let be a th-order character ; if is not a perfect th power , then we have the estimate where “” constant depends only on the degree of . Some related results can also be found in [7–10].

Now we are concerned about whether there exists an asymptotic formula for the hybrid mean value

In this paper, we will use the analytic method and the properties of character sums to study this problem and give a sharp asymptotic formula for (5) with , an odd prime. That is, we will prove the following.

Theorem 1. Let be an odd prime, let be any nonprincipal even character , and let be the principal character . Then we have the asymptotic formula where satisfies the inequalities .

From this theorem we may immediately deduce the following.

Corollary 2. For any odd prime and any nonprincipal even character with , one has
In the theorem, we only consider the polynomial . For general polynomial with and , whether there exists an asymptotic formula is complex problem for (5), it needs us to further study.
For general positive integer , whether there exists an asymptotic formula for (5) is also an interesting open problem.

2. Several Lemmas

To complete the proof of our theorem, we need the following several lemmas.

Lemma 1. Let be an odd prime and let be any nonprincipal even character . Then for any integer with , the identity where denotes the Legendre symbol and .

Proof. Since , there exists one and only one character such that . Thus, from the properties of Gauss sums we have
This proves Lemma 1.

Lemma 2. Let be an odd prime, let be any nonprincipal even character , , and , the principal character . Then for any integer and any quadratic nonresidue with , we have the identity

Proof. From the properties of Gauss sums we have
So from (11) we have
Note that and ; from (12) and Lemma 1 we may immediately deduce the identity
This proves Lemma 2.

Lemma 3. Let be an odd prime, let be any nonprincipal even character , , and , the principal character . Then for any integer and any quadratic nonresidue with , one has the estimate

Proof. Let be any integer such that or . Then from Lemma 2 we have
Note that ; applying (15) and Lemma 2 we have the estimate
This proves Lemma 3.

Lemma 4. Let be a prime. Then we have the identity where denotes the Legendre symbol.

Proof. For any odd prime and integer with , from Hua's book [11] (Section 7.8, Theorem 8.2) we know that
From this identity and the definition and properties of Gauss sums we have
This proves Lemma 4.

3. Proof of the Theorem

In this section, we will complete the proof of our theorem. Note that the identities and

So from (20), Lemmas 2, 3, and 4, and noting that we have where satisfies the inequalities .

This completes the proof of our theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the NSF (11371291) of China.