On the Hybrid Mean Value Involving Kloosterman Sums and Sums Analogous to Dedekind Sums

and Applied Analysis 3 = χ (−1) ⋅ τ 3 (χ) ⋅ 1 τ (χ) ⋅ p−1


Introduction
Let  be a natural number and let  be an integer prime to .The classical Dedekind sums where describe the behaviour of the logarithm of the eta-function (see [1,2]) under modular transformations.Gandhi [3] also introduced another sum analogous to Dedekind sums (ℎ, ) as follows: where  denotes any positive even number and ℎ denotes any integer with (ℎ, ) = 1.About the arithmetical properties of  2 (ℎ, ) and related sums, many authors had studied them and obtained a series of interesting results; see [1][2][3][4][5][6][7][8][9].For example, the second author [7] proved the following conclusion.
The sum  2 (ℎ, ) is important, because it has close relations with the classical Dedekind sums (ℎ, ).But unfortunately, so far, we knew that all results of  2 (ℎ, ) are the properties of their own, or the relationships between  2 (ℎ, ) and (ℎ, ), and had nothing to do with the other arithmetic functions.If we can find some relations between  2 (ℎ, ) and other arithmetic function, that will be very useful for further study of the properties of  2 (ℎ, ).
On the other hand, we introduce the classical Kloosterman sums (, ), which are defined as follows.For any positive integer  > 1 and integer , where  denotes the solution of the congruence  ⋅  ≡ 1 mod  and () =  2 .Some elementary properties of (, ) can be found in [10,11].
The main purpose of this paper is using the properties of the Gauss sums and the mean square value theorem of Dirichlet -functions to study a hybrid mean value problem involving  2 (ℎ, ) and Kloosterman sums and give two exact computational formulae for them.That is, we will prove the following.
Theorem 2. Let  be an odd prime; then one has the identity where ℎ  denotes the class number of the quadratic field Q(√−).

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 Gauss sums, all of which can be found in [12], so they will not be repeated here.First we have the following.
Lemma 3. Let  be an odd prime; then one has the identity Proof.It is clear that if  pass through a complete residue system mod, then 2 − 1 also pass through a complete residue system mod.So for any nonprincipal character  mod , from the properties of Gauss sums () (see Theorem 8.9 of [12]) we have the identity This proves Lemma 3.
Lemma 4. Let  > 2 be an integer; then for any integer  with (, ) = 1, one has the identity where (1, ) denotes the Dirichlet -function corresponding to character  mod .
Proof.Note that the divisors of 2 are 1, 2, , and 2.So from Lemma 4 and the definition of  2 (ℎ, 2) and (ℎ, ) we have where  denotes the principal character mod 2.
From the Euler infinite product formula (see Theorem where ∏  denotes the product over all primes .
From Lemma 4 we also have the identity Note that ℎ is an odd number; combining ( 14), (15), and ( 16) we have the identity This proves Lemma 5.

Lemma 6.
Let  be an odd prime.Then one has the identities Proof.From the definition of Dedekind sums we have If  ≡ 1 mod , then, from (20) and noting that the reciprocity theorem of Dedekind sums (see [5]), we have the computational formula 12 . ( Now taking  = 1 in (21), from ( 16) we may immediately deduce the identity Taking  = 2 in (21), from ( 16) we can also deduce the identity Now Lemma 6 follows from ( 22) and (23).

Proof of the Theorems
In this section, we will complete the proof of our theorems.(24) From (24) and Lemmas 4, 5, and 6 we have