Retracted Article: On the fourth power mean of the two-term exponential sums
© Zhu and Han; licensee Springer 2014
Received: 7 February 2014
Accepted: 6 March 2014
Published: 19 March 2014
The Retraction Note to this article has been published in Journal of Inequalities and Applications 2014 2014:344
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.
Keywordsthe two-term exponential sums fourth power mean Gauss sums identity asymptotic formula
where denotes the number of all distinct prime divisors of q.
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.
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.
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.
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.
where denotes the classical Gauss sum.
Now note that , if . From (2.1) and (2.2) we may immediately deduce Lemma 1. □
This proves Lemma 2. □
3 Proof of the theorem
Now our theorem follows from (3.4) and (3.6).
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).
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