Abstract
We use the analytic methods and the properties of Gauss sums to study the computational problem of one kind hybrid mean value involving the general Dedekind sums and the two-term exponential sums, and give an interesting computational formula for it.
1. Introduction
Let be a natural number and an integer prime to . The classical Dedekind sums where describes the behaviour of the logarithm of the eta-function (see [1, 2]) under modular transformations. About the various arithmetical properties of , many people had studied it and obtained a series of interesting results; see [3–9].
For example, Wang and Zhang [6] and Wang and Pan [7] had studied the hybrid mean value involving Dedekind sums and two-term exponential sums and proved the computational formulae where denotes the class number of the quadratic field , denotes the solution of the congruence equation , and the two-term exponential sums are defined as . Some results related to can be found in [10, 11].
On the other hand, Zhang [12] introduced a generalized Dedekind sums as follows: where denotes the th Bernoulli polynomial and defined for all real is called the th Bernoulli periodic function.
If , then , the classical Dedekind sums. About the arithmetical properties of and , one can find them in [3, 12]. In this paper as a note of [6, 7], we consider the following hybrid mean value: and use the analytic methods and the properties of Gauss sums to give an exact computational formula for (7). That is, we will prove the following conclusion.
Theorem 1. Let be a prime and any positive integer. Then for any positive integers and with and integer with , one has the identity For , , and , from our theorem we may immediately deduce the following.
Corollary 2. Let be a prime. Then for any positive integers and with and integer with , one has the identity
Corollary 3. Let be a prime. Then for any positive integers and with and integer with , one has the identity
Corollary 4. Let be a prime. Then for any positive integers and with and integer with , one has the identity
For general integer , whether there exists an exact computational formula for the hybrid mean value
is an open problem, where and are positive integers with and .
2. Several Lemmas
In this section, we will give two lemmas, which are necessary in the proof of our theorem. Hereinafter, we will use many properties of character sums and Gauss sums; all of these can be found in [13], so they will not be repeated here. First we have the following.
Lemma 1. Let be an odd prime and any nonprincipal character . Then for any positive integers and with and any integer , one has the identity where denotes the Gauss sums.
Proof. From the definitions of and Gauss sums we have
Since , then there exits one integer such that and . From the properties of reduced residue system we know that if pass through a reduced residue system , then also pass through a reduced residue system . So from (14) and Fermat little theorem we have
This proves Lemma 1.
Lemma 2. Let be an integer and any integer with . Then for any positive integer , one has the following identities.(i)If is an odd number, then (ii)If is an even number, then where denotes the Dirichlet -function corresponding to character and is the famous Riemann zeta-function.
Proof. See [12].
3. Proof of the Theorem
In this section, we will complete the proof of our theorem. If is an odd number, then note that for any nonprincipal character , and . So for any integer with , from (i) of Lemma 2 we have If is an even number, then note that and the identity From (ii) of Lemma 2 and the method of proving (18) we have Combining (18) and (20) we may immediately complete the proof of our theorem.
From the definition of we have Combining (18) and (21) we can deduce the identity
This proves Corollary 2.
If , then note that ; we have From (20) and (23) we have the identity This proves Corollary 3.
If , then note that ; we have From (18) and (23) we have the identity
This completes the proof of Corollary 4.
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 his 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.