A new sum analogous to quadratic Gauss sums and its 2k th power mean
© Xiancun and Xiaoxue; licensee Springer. 2014
Received: 29 January 2014
Accepted: 18 February 2014
Published: 4 March 2014
The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of a new sum analogous to quadratic Gauss sums, and to give an interesting asymptotic formula for its 2k th power mean.
Keywordsa sum analogous to quadratic Gauss sums 2k th power mean asymptotic formula
These two sums play a very important role in the study of analytic number theory, and many famous number theoretic problems are closely related to them. The distribution of primes, the Goldbach problem, and the properties of Dirichlet L-functions are some good examples. The arithmetic properties of and can be found in [1, 2], and .
where c, m, and n are any integers, χ is a non-principal Dirichlet character modq.
In this paper, we shall study the asymptotic properties of . As regards this problem, it seems that none has studied it yet, at least we have not seen any related results before. The problem is interesting, because it has a close relation with the Gauss sums, and it is also analogous to quadratic Gauss sums. Of course, it can also help us to further understand and study the quadratic Gauss sums.
The main purpose of this paper is using the analytic method and the properties of Gauss sums to study the 2k th power mean of , and to give a sharp asymptotic formula for it. That is, we shall prove the following two conclusions.
From Theorem 2 we can also deduce the following two corollaries.
is an interesting open problem, where m and n are any integers with .
2 Proof of the theorems
To complete the proofs of our theorems, we need the following simple conclusion.
where denotes the Legendre symbol.
where denotes the solution of the congruence equation .
For any non-principal character , note that ; from (2) we may immediately complete the proof of our lemma. □
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.S.F. (2013JZ001) 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
- Weil A: On some exponential sums. Proc. Natl. Acad. Sci. USA 1948, 34: 204-207. 10.1073/pnas.34.5.204MathSciNetView ArticleMATHGoogle Scholar
- Wenpeng Z: On the fourth power mean of the general Kloosterman sums. Indian J. Pure Appl. Math. 2004, 35: 237-242.MathSciNetMATHGoogle 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
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