Abstract and Applied Analysis

Volume 2014 (2014), Article ID 474726, 4 pages

http://dx.doi.org/10.1155/2014/474726

Research Article

On the Sixth Power Mean Value of the Generalized Three-Term Exponential Sums

Yahui Yu¹ and Wenpeng Zhang²

¹Luoyang Institute of Science and Technology, Luoyang, Henan 471023, China

²School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, China

Received 20 January 2014; Accepted 18 March 2014; Published 9 April 2014

Academic Editor: Shanhe Wu

Copyright © 2014 Yahui Yu and Wenpeng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main purpose of this paper is using the estimate for trigonometric sums and the properties of the congruence equations to study the computational problem of one kind sixth power mean value of the generalized three-term exponential sums and give an exact computational formula for it.

1. Introduction

Let be a positive integer and let be any Dirichlet character . For any integers and , the generalized three-term exponential sum is defined as follows:

where is a fixed integer and .

Many authors have studied this and related exponential sums and obtained a series of results; some related contents can be found in [1–9]. For example, Du and Han [5] proved that, for any integer , we have the identity

where the constant is defined as follows:

In particular, if , then we have the identity

It seems that no one has studied the sixth power mean of the generalized three-term exponential sums

where denotes the summation over all characters . The problem is interesting, because it can reflect more or less the upper bound estimates of . It is easy to see that mean value (2) is the best possible. So, we have reason to believe that (5) and (2) have similar asymptotic properties. In fact, we can use the analytic method and the properties of the congruence equation to give an exact computational formula for (5). That is, we will prove the following.

Theorem 1. Let be a prime. Then, for any integer with , we have the identity

It is very strange that the mean value in our theorem is independent of the size of ; it depends only on whether or . For any Dirichlet character , whether there exists a computational formula for the sixth power mean value

or -th () power mean value

is two open problems, which we will further study.

2. Proof of Theorem 1

In this section, we will give the proof of our theorem directly. Hereinafter, we will use many properties of trigonometric sums and congruence equation, all of which can be found in [6, 10], so they will not be repeated here. Note that , from the trigonometric identity

Regarding the properties of reduced residue system and the orthogonality relation for characters , we have

Now, we compute the values of and in (10), respectively. It is clear that the value of is equal to the number of the solutions of the system of the congruence equations:

where .

The system of congruence equation (11) is equivalent to the system of the congruence equations:

That is equivalent to the system of the congruence equations as follows:

For all integers , we compute the number of the solutions of (14). We separate the solutions of (14) into three cases as follows:

(A), ; , ; , ,(B), ; , ; , ,(C).

In case (C), (14) becomes and , so . That is, in this case, (14) has only one solution .

In case (B), if and , then (14) becomes , , , and or and , with . In this case, the number of the solutions of the congruence equation is . So, in case (B), the number of all solutions of congruence equation (14) is .

It is clear that the number of the solutions of congruence equation (14) in case (A) is three times the number of the solutions of the congruence equation , with and . While the number of the solutions of the latter congruence equation is . In fact, the congruence equation , with is equivalent to congruence equation , , . So, from (B), we know that the number of the solutions is . From (B) and (C), we know that the number of the solutions of congruence equation (14) in case (A) is .

Combining three cases (A), (B), and (C), we deduce that the number of all solutions of (14) is

Note that for all integers , the number of all solutions of the congruence equation

is also . In fact, all solutions of (16) are the solutions of (11). Thus, they are also the solutions of (14). On the other hand, any solution in (14) must belong to case (A), (B), or (C). From the computational process of the solutions in these three cases, we can see that any solution must satisfy (16). So, the number of the solutions of congruence equation (16) is .

Now, from (10), (11), (14), (15), and (16), we may immediately deduce the identity

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 referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P.S.F. (2013JZ001) and N.S.F. (11371291) of China.

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