An hybrid mean value of quadratic Gauss sums and a sum analogous to Kloosterman sums
© Pan and Zhang; licensee Springer. 2014
Received: 12 February 2014
Accepted: 17 March 2014
Published: 28 March 2014
The main purpose of this paper is, using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving the quadratic Gauss sums and a new sum analogous to Kloosterman sums, and to give an interesting hybrid mean value formula for it.
This sum plays a very important role in the study of analytic number theory, many famous number theoretic problems are closely related to it. For example, the distribution of primes, the Goldbach problem, the properties of Dirichlet L-functions are some good examples. About the arithmetic properties of , some authors had studied it and obtained many interesting results. For example, if is a prime and , then one can get the estimate . Some other results can be found in references [1–6].
where denotes the summation over all such that , and denotes the solution of the congruence equation .
The main purpose of this paper is using the analytic method and the properties of the character sums to study the hybrid mean value properties of and , and to give an interesting mean value formula. That is, we shall prove the following two conclusions.
where denotes the Legendre symbol, and denotes the classical Gauss sums with .
where denotes the summation over all even character , i.e. .
Some notes: Theorem 1 tells us that there exists a close relationship between and . That is, can be represented by .
Since for any odd character , we have , we only discussed the summation for all even characters in Theorem 2.
If , then we cannot give a computational formula for the hybrid mean value in Theorem 2. In this case, the difficulty is that we cannot obtain an exact value for the behind formula (13). We hope that the interested reader will stay with us as we turn to further study.
is an interesting open problem, where n is any integer with .
2 Several lemmas
In this section, we shall give two simple lemmas, which are necessary in the proofs of our theorems. Hereinafter, we shall use many properties of character sums and Gauss sums, all of these can be found in references [1, 2] and . First we have the following.
where denotes the Legendre symbol.
(This formula can be found in Hua’s book , Section 7.8, Theorem 8.2.)
Now Lemma 1 follows from (3) and (6). □
where denotes the Legendre symbol with .
This completes the proof of Lemma 2. □
3 Proof of the theorems
This proves Theorem 1.
This completes the proof of our theorems.
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.E.D. (2013JK0561) and N.S.F. (11371291) of P.R. China.
- Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.MATHGoogle Scholar
- Chengdong P, Chengbiao P: Goldbach Conjecture. Science Press, Beijing; 1992.MATHGoogle Scholar
- Ireland K, Rosen M: A Classical Introduction to Modern Number Theory. Springer, New York; 1982:204-207.View ArticleMATHGoogle Scholar
- Cochrane T, Pinner C: A further refinement of Mordell’s bound on exponential sums. Acta Arith. 2005, 116: 35-41. 10.4064/aa116-1-4MathSciNetView ArticleMATHGoogle Scholar
- Evans JW, Gragg WB, LeVeque RJ: On least squares exponential sum approximation with positive coefficients. Math. Comput. 1980,34(149):203-211. 10.1090/S0025-5718-1980-0551298-6MathSciNetView ArticleMATHGoogle Scholar
- Williams KS:Exponential sums over . Pac. J. Math. 1972, 40: 511-519. 10.2140/pjm.1972.40.511View ArticleMathSciNetMATHGoogle Scholar
- Burgess DA: On Dirichlet characters of polynomials. Proc. Lond. Math. Soc. 1963, 13: 537-548.MathSciNetView ArticleMATHGoogle Scholar
- Granville A, Soundararajan K: Large character sums: pretentious characters and the Pólya-Vinogradov theorem. J. Am. Math. Soc. 2007, 20: 357-384. 10.1090/S0894-0347-06-00536-4MathSciNetView ArticleMATHGoogle Scholar
- Zhang W, Yi Y: On Dirichlet characters of polynomials. Bull. Lond. Math. Soc. 2002, 34: 469-473. 10.1112/S0024609302001030MathSciNetView ArticleMATHGoogle Scholar
- Zhang W, Yao W: A note on the Dirichlet characters of polynomials. Acta Arith. 2004, 115: 225-229. 10.4064/aa115-3-3MathSciNetView ArticleMATHGoogle Scholar
- Hua LK: Introduction to Number Theory. Science Press, Beijing; 1979.Google Scholar
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