Abstract
The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the computational problem of one kind of new sum analogous to Gauss sums and give an interesting fourth power mean and a sharp upper bound estimate for it.
1. Introduction
Let be an integer, and let be a Dirichlet character . Then for any integer , famous Gauss sum is defined as follows: where .
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, Goldbach problem, the estimate of character sums, and the properties of Dirichlet -functions are some good examples.
It is clear that if is a primitive Dirichlet character , then we have and . Many properties of and can be found in [1–3].
In this paper, we introduce a new sum analogous to Gauss sums as follows: where is a character , and , , and are any integers with .
The main purpose of this paper is to study the properties of , such as the following two problems:(A)giving an upper bound estimate of ;(B)the problem of whether there exists a computational formula for the th power mean
It seems that no one has studied these problems yet; at least we have not seen any related results in the existing literature. The problems are interesting, because there exists a close relationship between the sum and the generalized Kloosterman sums; they can also help us to further understand the properties of hybrid mean value of an exponential sum and character of a polynomial.
For general integer , these two problems seem to be hard to make progress. But if is a prime and , then we can prove the following two conclusions.
Theorem 1. Let be an odd prime and any nonprincipal character . Then for any integers , , and with , one has the estimate
Theorem 2. Let be an odd prime and any nonprincipal character . Then for any integers and with , one has the identity
Some Notes. If is the principal character , then note that or and . So, in this case, the result is trivial; we do not need to discuss problems (A) and (B).
For any integer , whether there exists an exact computational formula for the th power mean is an open problem.
Furthermore, for general integer , whether there exists a nontrivial upper bound estimate for is also an interesting problem.
2. Several Lemmas
In this section, we will give several lemmas, which are necessary in the proof of our theorems. Hereinafter, we will use many properties of character sums and Gauss sums; all of these can be found in [1, 4]. So they will not be repeated here. First we have the following.
Lemma 3. Let be an odd prime; then, for any integer with , one has the identity where denotes the Legendre symbol.
Proof. From the properties of Gauss sums and quadratic residue we have This proves Lemma 3.
Lemma 4. Let be an odd prime and any nonprincipal Dirichlet character . Then for any integers and , one has the identity where denotes the solution of the congruence equation .
Proof. Since is a nonprincipal Dirichlet character , from Lemma 3 and the properties of Gauss sums we have For any nonprincipal character , we have . So, from (10) and noting that , we have This proves Lemma 4.
Lemma 5. Let be an odd prime and any Dirichlet character . Then for any integers and , one has the estimate where denotes the greatest common divisor of , , and .
Proof. This estimate is by Weil [5], Chowla [6], Malyshev [7], and Estermann [8] with some minor modifications.
Lemma 6. Let be an odd prime; then, for any integer with , one has the calculating formula
Proof. See [9] or Corollary 2 of [10].
3. Proof of the Theorems
In this section, we will complete the proof of our theorems. First note that if passes through a reduced residue system , then also passes through a reduced residue system , if . From these properties we have So without loss of generality we can assume that . Now we prove Theorem 1. From Lemmas 4 and 5 we have This proves Theorem 1.
Now we prove Theorem 2. From the properties of a complete residue system we know that if passes through a complete residue system , then also passes through a complete residue system . So from Lemma 4 we have If is the Legendre symbol, then is the principal character , so from (16) and Lemma 6 we have If is not a real character , then from (16) and Lemma 6 we have Now combining (14), (17), and (18) we may immediately deduce the identity This completes the proof of Theorem 2.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referee for the 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 China.