On the identity involving certain Hardy sums and Kloosterman sums
© Zhang and Zhang; licensee Springer. 2014
Received: 12 December 2013
Accepted: 13 January 2014
Published: 31 January 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.
KeywordsGauss sums Kloosterman sums identity certain Hardy sums hybrid mean value computational formula
where is the Riemann zeta-function and .
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.
where denotes the class number of the quadratic field .
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.
This proves Lemma 1. □
where denotes the Dirichlet L-function corresponding to character .
Proof See Lemma 2 of . □
Proof This formula is an immediate consequence of (5.9) and (5.10) in . □
where λ denotes the principal character .
where denotes the product over all primes p.
This proves Lemma 4. □
Now Lemma 5 follows from (8)-(11). □
3 Proof of the theorems
It is clear that Theorem 1 follows from (13) and (14).
Note that if , and if , then from (15) and (16) we may immediately deduce Theorem 2. This completes the proof of our theorems.
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.
- Rademacher H: On the transformation of . J. Indian Math. Soc. 1955, 19: 25-30.MathSciNetGoogle Scholar
- Rademacher H Carus Mathematical Monographs. In Dedekind Sums. Math. Assoc. of America, Washington; 1972.Google Scholar
- Berndt BC: Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 1978, 303/304: 332-365.MathSciNetMATHGoogle Scholar
- Apostol TM: Modular Functions and Dirichlet Series in Number Theory. Springer, New York; 1976.View ArticleMATHGoogle Scholar
- Carlitz L: The reciprocity theorem of Dedekind sums. Pac. J. Math. 1953, 3: 513-522. 10.2140/pjm.1953.3.513View ArticleMathSciNetMATHGoogle Scholar
- Conrey JB, Fransen E, Klein R, Clayton S: Mean values of Dedekind sums. J. Number Theory 1996, 56: 214-226. 10.1006/jnth.1996.0014MathSciNetView ArticleMATHGoogle Scholar
- Sitaramachandraro R: Dedekind and Hardy sums. Acta Arith. 1987, 48: 325-340.MathSciNetGoogle Scholar
- Zhang W, Yi Y: On the 2 m -th power mean of certain Hardy sums. Soochow J. Math. 2000, 26: 73-84.MathSciNetMATHGoogle Scholar
- Zhang W: On the mean values of Dedekind sums. J. Théor. Nr. Bordx. 1996, 8: 429-442. 10.5802/jtnb.179View ArticleMathSciNetMATHGoogle Scholar
- Chowla S: On Kloosterman’s sum. Nor. Vidensk. Selsk. Fak. Frondheim 1967, 40: 70-72.MathSciNetMATHGoogle Scholar
- Malyshev AV: A generalization of Kloosterman sums and their estimates. Vestn. Leningr. Univ. 1960, 15: 59-75. (in Russian)MathSciNetGoogle Scholar
- Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.MATHGoogle Scholar
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