On the Hybrid Power Mean Involving the Character Sums and Dedekind Sums

which unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.


Introduction
We all know that the classical Dedekind sums S(h, q) is defined (see [1]) as where q ≥ 2 is a positive integer, h is any integer prime to q, and is sum describes the behaviour of the logarithm of the eta function under modular transformations, see [1,2], for related references. Because of the importance of this sum in analytic number theory, many scholars have studied its various properties and obtained a series of important results. Perhaps, the most important property of S(h, q) is its reciprocity theorem (see [3]). at is, for any positive integers h and q with (h, q) � 1, one has the identity Some other papers related to Dedekind sums can be found in [4][5][6], and we do not want to list them all here.
On the contrary, we also introduce another character sums analogous to Kloosterman sums as follows. For any integer q ≥ 3, let χ be a Dirichlet character mod q. For any positive integer k and integer h, we define where q a�1 ′ denotes the summation over all 1 ≤ a ≤ q such that (a, q) � 1 and a denotes the inverse of a.
at is, a · a ≡ 1 mod q.
About the properties of G(k, h, χ; q), some people had studied it and obtained some important results. For example, from the very special case of Weil's work [7] one can obtain the estimate where p is a prime and m is any integer. Some related important works can also be found in [7][8][9][10][11].
In this paper, we consider the computational problem of the hybrid power mean involving the Dedekind sums S(h, q) and G(k, h, χ; q). at is, However, for this hybrid power mean, it seems that none has studied it yet; at least, we have not seen any related results before. e problem is interesting because it is closely related to Dirichlet L-functions. In fact, for some special positive integers k, we can give an exact computational formula for (6). e main work of this paper is to reveal this point.
at is, we shall use the elementary and analytic methods, and the properties of character sums to prove the following two conclusions. Theorem 1. Let p be an odd prime with p ≡ 3 mod 4, and χ 2 � ( * /p) denotes Legendre's symbol mod p. en, for any positive integer k with (2k + 1, p − 1) � 1, we have the identity where h p denotes the class number of the quadratic field Theorem 2. Let p be an odd prime with p ≡ 1 mod 8 and k be any positive integer with (k, p − 1) � 1. en, we have the identity If p be an odd prime with p ≡ 5 mod 8, then we have the identity where χ 4 denotes the fourth-order character mod p, α � (p− 1)/2 a�1 ((a + a)/p) is an integer, and L(s, χ) denotes the Dirichlet L-function corresponding to χ.
Taking k � 1 in eorem 1 and eorem 2, then we have the following.

Corollary 1.
Let p be an odd prime with p ≡ 3 mod 4 and (3, p − 1) � 1; then, we have the identity Corollary 2. Let p be an odd prime with p ≡ 5 mod 8; then, we have Notes: Obviously, in a sense, Corollary 1 gives us efficient methods to compute the class number h p that can be done on a computer.
It is easy to prove that if p ≡ 3 mod 4, then, for any positive integer k, we have If p ≡ 1 mod 4, then, for any positive integer k, we also have For general composite number q > 3, whether there exists an exact computational formula for (6) will be our further research problem.

Several Lemmas
In this section, we shall give several simple lemmas, and they are necessary in the proofs of our theorems. First, we have the following. Lemma 1. Let p > 3 be a prime, and λ and χ are two nonprincipal characters mod p with χ(− 1) � − 1. en, for any positive integer k, we have the identity where τ(χ) denotes the classical Gauss sums.
Proof. For any integer n and nonprincipal character χ mod p, from the properties of Gauss sums τ(χ) (see eorem 8.20 in [12]), we have Using (15) and the properties of the reduced residue system mod p, we have So, with the repeated use of (15) in (16), we have is proves Lemma 1.
□ Lemma 2. 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 character χ mod d.
□ Lemma 3. If p is a prime with p ≡ 1 mod 4 and ψ is any fourth-order character mod p, then we have the identity where α � (p− 1)/2 a�1 ((a + a)/p) is an integer.

Lemma 4.
If p is a prime with p ≡ 5 mod 8 and ψ is any fourth-order character mod p, then, for any positive integer k, we have the identity Proof. First, for all nonnegative integers u and real numbers X and Y, we have the identity where [x] denotes the greatest integer ≤x. is formula is obtained because of Waring [15]. It can also be found in [16].