On the General Dedekind Sums and Two-Term Exponential Sums

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.


Introduction
Let be a natural number and ℎ an integer prime to . The classical Dedekind sums describes the behaviour of the logarithm of the eta-function (see [1,2]) under modular transformations. About the various arithmetical properties of (ℎ, ), many people had studied it and obtained a series of interesting results; see [3][4][5][6][7][8][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 ℎ denotes the class number of the quadratic field Q(√− ), denotes the solution of the congruence equation ≡ 1 mod , and the two-term exponential sums ( , , ℎ, ; ) are defined as ( ) = 2 . Some results related to ( , , ℎ, ; ) can be found in [10,11].

2
The Scientific World Journal On the other hand, Zhang [12] introduced a generalized Dedekind sums as follows: where if is an integer, ( ) denotes the th Bernoulli polynomial and ( ) defined for all real 0< ≤ 1 is called the th Bernoulli periodic function.
If = 1, then (ℎ, 1; ) = (ℎ, ), the classical Dedekind sums. About the arithmetical properties of (ℎ, ; ) 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.  For = 1, 2, and 3, from our theorem we may immediately deduce the following.

Corollary 2. Let
≥ 3 be a prime. Then for any positive integers ℎ and with (ℎ , − 1) = 1 and integer with ( , ) = 1, one has the identity Corollary 4. Let ≥ 3 be a prime. Then for any positive integers ℎ and with (ℎ , − 1) = 1 and integer with ( , ) = 1, one has the identity For general integer > 3, whether there exists an exact computational formula for the hybrid mean value is an open problem, where ℎ and are positive integers with ( ℎ, ( )) = 1 and ( , ) = 1.

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. Proof. From the definitions of ( , , ℎ, ; ) and Gauss sums we have Since (ℎ , − 1) = 1, then there exits one integer such that ⋅ ≡ 1 mod ( − 1) and ( , − 1) = 1. From the properties of reduced residue system mod we know that if pass through a reduced residue system mod , then also This proves Lemma 1.

Lemma 2. Let ≥ 3 be an integer and ℎ any integer with
(ℎ, ) = 1. 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 (1, ) denotes the Dirichlet -function corresponding to character mod and ( ) is the famous Riemann zetafunction.

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 If ≥ 2 is an even number, then note that ( , − 1) = 1 and the identity From (ii) of Lemma 2 and the method of proving (18) we have The Scientific World Journal Combining (18) and (20) we may immediately complete the proof of our theorem. From the definition of (1, 1; ) we have Combining (18) and (21) we can deduce the identity This completes the proof of Corollary 4.