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Retracted Article: On the fourth power mean of the two-term exponential sums
Journal of Inequalities and Applications volume 2014, Article number: 116 (2014)
The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind of fourth power mean of two-term exponential sums, and to give an interesting identity and asymptotic formula for it.
Let be a positive integer. For any integers m and n, the two-term exponential sum is defined as follows:
About the various properties of , some authors had studied it, and they obtained a series of results, some related works can be found in references [1–5]. For example, Gauss’ classical work proved the remarkable formula (see )
Generally, for any odd number q and , the exact value of is (e.g. see Berndt, Evans, Williams and Apostol’s related works). Cochrane and Zheng  show for the general sum that
where denotes the number of all distinct prime divisors of q.
In this paper, we study the fourth power mean of the two-term exponential sum as follows:
where n is any integer with .
As regards this problem, it seems that none has yet studied it, at least we have not seen any related result before. The problem is interesting, because it can reflect that the mean value of is well behaved. The main purpose of this paper is to show this point. That is, we shall prove the following conclusion.
Theorem Let be a prime. Then for any integer n with , we have the identity
where is any 3-order character .
Note that for any non-principal character , we have , so from our theorem we may immediately deduce the following.
Corollary Let be a prime with . Then for any integer n with , we have the asymptotic formula
It seems that our method can also be used to deal with (1.1) for all prime p and integer . But this time, the computing is very complex.
For any integer , whether there exists an exact computational formula for
where p is an odd prime and , is an open problem.
2 Several lemmas
In this section, we will give several lemmas which are necessary in the proof of our theorem. In the proving process of all lemmas, we used many properties of Gauss sums; all these can be found in , we will not be repeat them here. First we have the following.
Lemma 1 Let p be an odd prime, χ be any non-principal character . Then for any integer n with , we have the identity
Proof Note that χ is a non-principal character , so if χ is not a 3-order character (that is, , the principal character ), then from the properties of Gauss sums we have
where denotes the classical Gauss sum.
If χ is a 3-order character , then ; note that for any integer a with , we have
From the method of proving (2.1) we have the identity
Now note that , if . From (2.1) and (2.2) we may immediately deduce Lemma 1. □
Lemma 2 Let p be an odd prime, χ be any non-real character . Then we have the identity
Proof From the definition and properties of the classical Gauss sums we have
This proves Lemma 2. □
3 Proof of the theorem
In this section, we shall complete the proof of our theorem. First from the orthogonality of characters we have
On the other hand, if , then any non-principal character χ is not a 3-order character . Note that
From (3.2) and Lemma 1 we have
If , then combining (3.1) and (3.3) we may immediately deduce the identity
If , let be a 3-order character , then is also a 3-order character ; this time note that
and from Lemma 2 and the method of proving (3.3) we have
So if , then combining (3.1) and (3.5) we can deduce the asymptotic formula
Now our theorem follows from (3.4) and (3.6).
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Cochrane T, Zheng Z: Bounds for certain exponential sums. Asian J. Math. 2000, 4: 757–774.
Cochrane T, Zheng Z: Upper bounds on a two-term exponential sums. Sci. China Ser. A 2001, 44: 1003–1015. 10.1007/BF02878976
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Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.
The authors would like to thank the referee for his 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 P.R. China, and the SF of Education of Shaanxi province (12JK0877).
The authors declare that they have no competing interests.
The main idea of this paper was proposed by MZ and DH. All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
An erratum to this article can be found at http://dx.doi.org/10.1186/1029-242X-2014-344.
Following publication of this article , it was brought to the attention of the editorial and publishing staff that an identically-titled article by Han Zhang and Wenpeng Zhang was published in January 2014 in The Scientific World Journal.
This is a violation of publication ethics which, in accordance with the Springer Policy on Publishing Integrity, warrants a retraction of the article and a notice to this effect to be published in the journal.
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Zhu, M., Han, D. Retracted Article: On the fourth power mean of the two-term exponential sums. J Inequal Appl 2014, 116 (2014). https://doi.org/10.1186/1029-242X-2014-116
- the two-term exponential sums
- fourth power mean
- Gauss sums
- asymptotic formula