A Hybrid Mean Value Involving Dedekind Sums and the Generalized Kloosterman Sums

In this paper, we use the mean value theorem of Dirichlet 
 
 L
 
 -functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.


Introduction
Let q be a natural number and h be an integer coprime to q. e classical Dedekind sums where describes the behaviour of the logarithm of the η-function (see [1,2]) under modular transformations. ere are many papers written on their various properties (see the examples in [3][4][5][6][7][8][9][10] and [11]). In particular, Zhang and Liu [12] studied the hybrid mean value problems related to Dedekind sums and Kloosterman sums: where q ≥ 3 is an integer, q′ a�1 denotes the summation over all 1 ≤ a ≤ q with (a, q) � 1, e(y) � e 2πiy , and a denotes the multiplicative inverse of a mod q. ey proved the following results: Theorem 1. Let p be an odd prime, then one has the identity where h p denotes the class number of the quadratic field Q( �� � − p √ ).
Theorem 2. Let p be an odd prime, then one has the asymptotic formula: where exp(y) � e y .
It is natural that people will ask, for the general Kloosterman sums what will happen? Whether there exists an identity similar to eorem1? Here, χ denotes any Dirichlet character mod p.
e main purpose of this paper is to answer these questions. at is, we shall use the mean value theorem of Dirichlet L-functions and the properties of Gauss sums and Dedekind sums to prove the following. Theorem 3. Let p be an odd prime with p ≡ 1 mod 4. en, for any Dirichlet character χ mod p, we have the identity: Theorem 4. Let p be an odd prime with p ≡ 3 mod 4. en, for any Dirichlet character χ mod p, we have the identity: where χ 2 � ( * /p) denotes the Legendre symbol and h p denotes the class number of the quadratic field Q( It is clear that if χ � χ 0 , then K(a, 1, χ; p) � K(a, 1; p). Note that χ 0 (− 1) � 1, from eorems 3 and 4, we may immediately deduce eorem1 in [12], so our results are the generalization of [12].

Several Lemmas
In this section, we shall give several simple lemmas, which are necessary to the proofs of our theorems. Hereafter, we shall use many properties of character sums and Gauss sums, and all of these can be found in reference [13]. First, we have the following.

Lemma 1. Let p > 3 be a prime, χ be any fixed Dirichlet character mod p.
en, for any nonprincipal character χ 1 mod p with χχ 1 ≠ χ 0 , we have the identity: where χ 0 denotes the principal character mod p, τ(χ) denotes the Gauss sums defined as τ(χ) � On the other hand, from the properties of Gauss sums, we have Combining (10) and (11), we may immediately deduce the identity: is proves Lemma 1.

Lemma 2.
Let p be a prime with p ≡ 1 mod 4 and χ be any odd character mod p. en, we have the identity: Proof. Since p ≡ 1 mod 4 and χ is an odd character mod p, we know χ is not the Legendre symbol and χ 2 ≠ χ 0 . Note that χ(− 1) � − 1, from (10), we have is proves Lemma 2.

Lemma 4.
Let q > 2 be an integer, then for any integer a with (a, q) � 1, we have the identity: where L(1, χ) denotes the Dirichlet L-function corresponding to the character χ mod d.

Proof of the Theorems
In this section, we will complete the proof of our theorems. First we prove eorem 3. From Lemma 4 and the definition of S(a, p), we have and (with a � 1) Since p ≡ 1 mod 4, we know the Legendre symbol ( * /p) � χ 2 is an even character mod p. Note that, for any nonprincipal character χ mod p, |τ(χ)| � � � p √ . So, if χ is an even character mod p, then from Lemma 1, (18), and (19), we have If χ is an odd character mod p, then note that the identity: · L 1, χ 1 2 + p · π − 2 p − 1 χ 1 mod p