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The fourth power mean of the generalized two-term exponential sums and its upper and lower bound estimates
Journal of Inequalities and Applicationsvolume 2013, Article number: 504 (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.
Let be a positive integer. For any integers m and n, the generalized two-term exponential sum is defined by
where χ denotes any Dirichlet character , and .
Regarding the upper bound estimate of , many authors have studied it and obtained a series of important results; related contents can be found in [1–5] and . For example, from Weil’s classical work  one can deduce the estimate
Recently, Wang  studied the computational problem of the fourth power mean of , and proved the following conclusion:
Let p be an odd prime with . Then, for any integer m with , we have the identity
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:
Let p be an odd prime. Then, for any non-principal even character and any character with , we have the asymptotic formula
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.
Theorem Let p be an odd prime. Then, for any character , we have the identity
where denotes the principal character , , and is the Legendre symbol.
From this theorem we may immediately deduce the following corollary.
Corollary Let p be an odd prime. Then, for any non-principal character , we have the inequalities
2 Several lemmas
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.
Lemma 1 Let p be an odd prime. Then, for any integers m and n with , we have the identity
where denotes the Legendre symbol, and .
Proof See Lemma 1 in . □
Lemma 2 Let p be an odd prime, be any fixed character . Then, for any non-real character , we have the identity
Proof From the properties of Gauss sums, we have
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. □
Lemma 3 Let p be an odd prime, χ be any non-principal character with . Then we have
Proof From the properties of quadratic residue , we have
so from (2) we may immediately deduce the identity
follows from Lemma 1 of . This proves Lemma 3. □
3 Proof of Theorem
In this section, we shall give two different proofs of our theorem. First, if is a non-principal character , then from Lemma 1 we have
where and (see Theorem 7.5.4 of ).
From (3) and the definition of Gauss sums, we may immediately deduce
If is a non-principal character with , then note that
From (4) we have
If is a non-principal character with , then from (4) and Lemma 3 we have
If is the principal character , then from the method of proving (3) and (4) we have
Combining (5), (6) and (7), we may immediately deduce our theorem.
The second proof of Theorem. First, from the orthogonality of characters , we have
On the other hand, from Lemma 2 we have
Applying the orthogonality of characters , we can easily deduce that
From the definition and properties of Gauss sums, we have
Note that if , then
Combining (7)-(14) and Lemma 3, we may immediately deduce the identity
This completes another proof of our theorem.
The corollary follows from Theorem and Lemma 3.
Cochrane T, Zheng Z: Upper bounds on a two-term exponential sums. Sci. China Ser. A 2001, 44: 1003–1015. 10.1007/BF02878976
Cochrane T, Zheng ZY: Pure and mixed exponential sums. Acta Arith. 1999, 91: 249–278.
Cochrane T, Pinner C: A further refinement of Mordell’s bound on exponential sums. Acta Arith. 2005, 116: 35–41. 10.4064/aa116-1-4
Cochrane T, Pinner C: Using Stepanov’s method for exponential sums involving rational functions. J. Number Theory 2006, 116: 270–292. 10.1016/j.jnt.2005.04.001
Hua LK: Introduction to Number Theory. Science Press, Beijing; 1979.
Zhang W: Moments of generalized quadratic Gauss sums weighted by L -functions. J. Number Theory 2002, 92: 304–314. 10.1006/jnth.2001.2715
Weil A: On some exponential sums. Proc. Natl. Acad. Sci. USA 1948, 34: 204–207. 10.1073/pnas.34.5.204
Wang T: On the fourth power mean of generalized two-term exponential sums. Bull. Korean Math. Soc. 2013, 50: 233–240. 10.4134/BKMS.2013.50.1.233
Wang J: Hybrid mean value of the generalized Kloosterman sums and Dirichlet character of polynomials. Bull. Korean Math. Soc. 2013, 50: 451–458. 10.4134/BKMS.2013.50.2.451
Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.
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.
The authors declare that they have no competing interests.
LX carried out the part of Introduction, XZ carried out the proof of some lemmas, LX with XZ carried out the theorem’s proof. All authors read and approved the final manuscript.