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On the identity involving certain Hardy sums and Kloosterman sums
Journal of Inequalities and Applications volume 2014, Article number: 52 (2014)
The main purpose of this paper is, using the properties of Gauss sums and the mean value theorem of Dirichlet L-functions, to study a hybrid mean value problem involving certain Hardy sums and Kloosterman sums and give two exact computational formulae for them.
Let c be a natural number and d be an integer prime to c. The classical Dedekind sums
describes the behavior of the logarithm of the eta-function (see  and ) under modular transformations. Berndt  gave an analogous transformation formula for the logarithm of the classical theta-function
That is, put with , , and . Then we have
where are defined as
The sums (and certain related ones) are sometimes called Hardy sums. They are closely connected with Dedekind sums. Some authors had studied the properties of and related sums and obtained some interesting results, see [4–8] and . For example, Zhang and Yi  proved the following conclusion. Let p be an odd prime. Then, for any fixed positive integer m, we have the asymptotic formula
where is the Riemann zeta-function and .
On the other hand, we introduce the classical Kloosterman sums which are defined as follows: For any positive integer and integer n,
The main purpose of this paper is, using the properties of Gauss sums and the mean square value theorem of Dirichlet L-functions, to study a hybrid mean value problem involving certain Hardy sums and Kloosterman sums and give two exact computational formulae. That is, we shall prove the following theorem.
Theorem 1 Let p be an odd prime. Then we have the identity
Theorem 2 Let p be an odd prime, then we have the identity
where denotes the class number of the quadratic field .
For general odd number , whether there exits a computational formula for the hybrid mean value
is an open problem.
2 Several lemmas
In this section, we shall give several lemmas, which are necessary in the proof of our theorems. Hereinafter, we shall use many properties of Gauss sums, all of which can be found in , so they will not be repeated here. First we have the following lemma.
Lemma 1 Let p be an odd prime, then we have the identity
Proof For any non-principal character , from the properties of Gauss sums we have
This proves Lemma 1. □
Lemma 2 Let be an integer, then, for any integer a with , we have the identity
where denotes the Dirichlet L-function corresponding to character .
Proof See Lemma 2 of . □
Lemma 3 Let and . Then we have the identity
Proof This formula is an immediate consequence of (5.9) and (5.10) in . □
Lemma 4 Let p be an odd prime and . Then we have the identity
Proof Note that the divisors of 2p are 1, 2, p and 2p. So from Lemma 2 and Lemma 3 we have
where λ denotes the principal character .
From the Euler infinite product formula we have
where denotes the product over all primes p.
From Lemma 2 we also have the identity
Now, combining (2), (3) and (4), we have the identity
This proves Lemma 4. □
Lemma 5 Let p be an odd prime. Then we have the identities
Proof From the definition of Dedekind sums we have
If , then from (5), and noting the reciprocity theorem of Dedekind sums (see ), we have the computational formula
If , then we also have
Now taking in (6), from (4) we may immediately deduce the identity
Taking in (6), from (4) we can also deduce the identity
If , then taking in (6), from (4) we can deduce the identity
If , then from (4) and (7) we have the identity
Now Lemma 5 follows from (8)-(11). □
3 Proof of the theorems
In this section, we shall complete the proof of our theorems. Note that if χ is a non-principal character , then and
If , then from (12), Lemma 4 and Lemma 5 we have
If , then from (12), Lemma 4 and Lemma 5 we also have
It is clear that Theorem 1 follows from (13) and (14).
Now we prove Theorem 2. If , then from Lemma 1 and the method of proving Theorem 1 we have
If , then note that the Legendre symbol , , and
so from Lemma 1 and the method of proving Theorem 1 we have
Note that if , and if , then from (15) and (16) we may immediately deduce Theorem 2. This completes the proof of our theorems.
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The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N.S.F. (11371291), the S.R.F.D.P. (20136101110014), the N.S.F. (2013JZ001) of Shaanxi Province, P.R. China.
The authors declare that they have no competing interests.
HZ obtained the theorems and completed the proof. WZ corrected and improved the final version. Both authors read and approved the final manuscript.
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Zhang, H., Zhang, W. On the identity involving certain Hardy sums and Kloosterman sums. J Inequal Appl 2014, 52 (2014). https://doi.org/10.1186/1029-242X-2014-52
- Gauss sums
- Kloosterman sums
- certain Hardy sums
- hybrid mean value
- computational formula