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A hybrid mean value involving a new sum and Kloosterman sums
Journal of Inequalities and Applications volume 2014, Article number: 44 (2014)
In this paper, we introduce a new sum, analogous to Cochrane sums, and use elementary and analytic methods to study the hybrid mean value problem involving this sum and Kloosterman sums, and we give an interesting asymptotic formula for it.
Let q be a natural number and h an integer with . The Cochrane sum is defined by
is defined by , and denotes the summation over all such that .
This sum was introduced by Professor Todd Cochrane, and it has been studied by many authors, and one obtained many interesting results. Related works can be found in [1–3] and [4–11]. For example, Zhang  studied the hybrid mean value properties of Cochrane sums and Kloosterman sums and proved that for any prime , we have the asymptotic formula
denotes the Kloosterman sum, and .
Now for any odd prime and integer h with , we define another sum analogous to the Cochrane sum as follows:
The main purpose of this paper is using elementary and analytic methods to study the hybrid mean value problem involving and the Kloosterman sum and give an asymptotic formula for it. That is, we shall prove the following.
Theorem Let p be an odd prime. Then we have the asymptotic formula
For general odd number , whether there exists an asymptotic formula for
is still an open problem.
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 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 integer a with , we have the identity
where χ runs through the Dirichlet characters modp with , and denotes the classical Gauss sum corresponding to χ.
Proof See reference . □
Lemma 2 Let p be an odd prime and h an integer with . We define as follows:
Then we have the identity
where denotes the Legendre symbol.
Proof From the orthogonality relation for characters modp we have
If is the principal character modp, then we have
If χ is an even character modp (that is, χ is a non-principal character and ), then we have
From this identity, (2), and (3) we know that if χ is an even character modp, then
If χ is an odd character modp and . Let denote the Legendre symbol, then must be an even character modp, so we have
If χ is an odd character modp and , then must be an odd character modp, so we have
Combining (1), (2), (4)-(7) we may immediately deduce the identity
This proves Lemma 2. □
Lemma 3 Let be an odd prime, then we have the asymptotic formula
Proof For any non-principal character , applying Abel’s identity (see Theorem 4.2 of ) we have
where denotes the divisor function, and .
From  we know that for any real number , we have the estimate
From (8) and (9) we can deduce that
where we have used the estimate , is any fixed real number. This proves Lemma 3. □
Lemma 4 Let be a prime with . Then we have the estimates
Proof From the method of proving Lemma 3 we have
So applying (10) we may immediately deduce the estimate
This proves Lemma 4. □
3 Proof of the theorem
In this section, we shall use the lemmas to complete the proof of our theorem. For any prime , note that the identities
If , then from Lemma 1, Lemma 2, and the properties of the Gauss sum we have
Note that and that we have the identity
so from Lemma 4 we have
Combining (11), (13), and Lemma 3 we can deduce the asymptotic formula
If , then from Lemma 2 we know that , so from Lemma 3 and the method of proving (14) we also have
Now the theorem follows from the asymptotic formulas (14) and (15).
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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 P.R. China.
The authors declare that they have no competing interests.
XW carried out the proofs of the lemmas, XL carried out the part of Introduction, XW and XL carried out the theorem’s proof. All authors read and approved the final manuscript.
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Wang, X., Li, X. A hybrid mean value involving a new sum and Kloosterman sums. J Inequal Appl 2014, 44 (2014). https://doi.org/10.1186/1029-242X-2014-44
- new sum analogous to Cochrane sums
- Kloosterman sums
- hybrid mean value
- asymptotic formula