- Open Access
The fourth power mean of the generalized two-term exponential sums and its upper and lower bound estimates
© Li and Xu; licensee Springer. 2013
- Received: 13 June 2013
- Accepted: 2 October 2013
- Published: 8 November 2013
In this paper, we use the analytic method and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the generalized two-term exponential sums, and give an exact computational formula for it.
- generalized two-term exponential sums
- fourth power mean
- computational formula
where χ denotes any Dirichlet character , and .
Recently, Wang  studied the computational problem of the fourth power mean of , and proved the following conclusion:
Wang  studied the hybrid power mean of the generalized Kloosterman sums and , where λ denotes a Dirichlet character , and gave an interesting asymptotic formula for it. That is, she proved the following result:
In this paper, as a note of  and , we found that there exists a close relationship between the fourth power mean of and . The main purpose of this paper is to show this point. That is, we shall prove the following theorem.
where denotes the principal character , , and is the Legendre symbol.
From this theorem we may immediately deduce the following corollary.
In this section, we shall give several lemmas, which are necessary in the proof of our theorem. Hereinafter, we shall use many properties of character sums and Gauss sums, all of these can be found in reference , so they will not be repeated here. First we have the following.
where denotes the Legendre symbol, and .
Proof See Lemma 1 in . □
Note that , χ is a non-real character , so is also a non-principal character . Therefore, , so from (1) we may immediately deduce Lemma 2. □
follows from Lemma 1 of . This proves Lemma 3. □
where and (see Theorem 7.5.4 of ).
Combining (5), (6) and (7), we may immediately deduce our theorem.
This completes another proof of our theorem.
The corollary follows from Theorem and Lemma 3.
The authors would like to thank the referee for his/her very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N.S.F. (11071194) of P.R. China.
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