A hybrid mean value involving a new sum and Kloosterman sums
© Wang and Li; licensee Springer. 2014
Received: 12 December 2013
Accepted: 10 January 2014
Published: 27 January 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.
Keywordsnew sum analogous to Cochrane sums Kloosterman sums hybrid mean value asymptotic formula
is defined by , and denotes the summation over all such that .
denotes the Kloosterman sum, and .
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.
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.
where χ runs through the Dirichlet characters modp with , and denotes the classical Gauss sum corresponding to χ.
Proof See reference . □
where denotes the Legendre symbol.
This proves Lemma 2. □
where denotes the divisor function, and .
where we have used the estimate , is any fixed real number. This proves Lemma 3. □
This proves Lemma 4. □
3 Proof of the theorem
Now the theorem follows from the asymptotic formulas (14) and (15).
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.
- 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
- Carlitz L: The reciprocity theorem of Dedekind sums. Pac. J. Math. 1953, 3: 513-522. 10.2140/pjm.1953.3.513View ArticleMathSciNetMATHGoogle Scholar
- Jia CH: On the mean value of Dedekind sums. J. Number Theory 2001, 87: 173-188. 10.1006/jnth.2000.2580MathSciNetView ArticleMATHGoogle Scholar
- Conrey JB, Fransen E, Klein R, Scott C: Mean values of Dedekind sums. J. Number Theory 1996, 56: 214-226. 10.1006/jnth.1996.0014MathSciNetView ArticleMATHGoogle Scholar
- Gandhi JM: On sums analogous to Dedekind sums. Proceedings of the Fifth Manitoba Conference on Numerical MathematicsWinnipeg, Manitoba, 1975, 1976, 647-655.Google Scholar
- 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
- Zhang WP: A note on the mean square value of the Dedekind sums. Acta Math. Hung. 2000, 86: 275-289. 10.1023/A:1006724724840View ArticleMathSciNetMATHGoogle Scholar
- Zhang WP: On the mean values of Dedekind sums. J. Théor. Nr. Bordx. 1996, 8: 429-442. 10.5802/jtnb.179View ArticleMathSciNetMATHGoogle Scholar
- Zhang WP: A sum analogous to Dedekind sums and its hybrid mean value formula. Acta Arith. 2003, 107: 1-8. 10.4064/aa107-1-1MathSciNetView ArticleMATHGoogle Scholar
- Zhang WP: On a Cochrane sum and its hybrid mean value formula. J. Math. Anal. Appl. 2002, 267: 89-96. 10.1006/jmaa.2001.7752MathSciNetView ArticleMATHGoogle Scholar
- Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.