Some Identities Involving Certain Hardy Sums and General Kloosterman Sums

Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l , λ ; p ) and certain Hardy sums S 1 ( h , q ) ∑ m = 1 p − 1 ∑ s = 1 p − 1 K ( m , n , λ ; p ) K ( s , t , λ ; p ) S 1 ( 2 m s ¯ , p ) , ∑ m = 1 p − 1 ∑ s = 1 p − 1 | K ( m , n , λ ; p ) | 2 | K ( s , t , λ ; p ) | 2 S 1 ( 2 m s ¯ , p ) . Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character λ modulo p, while we consider all the cases.


Introduction
Let k be a positive integer and h be arbitrary integer with (h, k) = 1, the classical Dedekind sums S(h, k) are defined as S(h, k) also can be written as which belong to the family of "cotangent sums". These sums have a wide range of applications and in some cases relations to some major open problems, such as Riemann Hypothesis [1][2][3][4] So the cotangent sums arise in connection with the Nyman-Beurling approach to the Riemann Hypothesis.
Dedekind sums also play an important role in the transformation theory of the Dedekind η function. Dedekind sums have many interesting properties. For example, L.Carlitz [5] obtained the reciprocity theorem of S(h, k) as W. P. Zhang [6] established the relationship between Dedekind sums and Dirichlet L-function: where k > 2 is an integer, a is an integer with (a, k) = 1, ϕ(d) denotes the Euler function and L(1, χ) denotes the Dirichlet L-function corresponding to character χ modulo d. Some scholars also studied Dedekind type sums and obtained interesting results. For example, by considering Dedekind type DC(Daehee-Changhee) sums where E p (x) are the p-th Euler functions, T. Kim [7] proved reciprocity law: is the largest integer ≤ x, E n are the n-th Euler numbers. Later T. Kim [8] defined p-adic Dedekind-type DC sums as follows: where (h, k) = (p, k) = 1, s ∈ Z p , and T q (m, a, N : q N ) is a continuous p-adic extension of , then he got a continuous function S p,q (m : h, k : q k ) on Z p , which satisfies The other sums analogous to Dedekind sums are defined as where h and k are integers with k > 0. The sums S 1 (h, k) are sometimes called Hardy sums. Some authors studied the properties of S 1 (h, k) and related sums, and obtained some interesting results, see [9][10][11][12]. A relation between certain Hardy sums S 1 (h, k) and classical Dedekind sums S(h, k) can be obtained in [12] that if (h, k) = 1, then Other scholars showed their interests to the hybrid mean value involving Hardy sums and other famous sums, see [13][14][15]. For example, W. P. Zhang [15] studied the hybrid mean value involving certain Hardy sums S 1 (h, k) and Kloosterman sums denotes the summation over all a with (a, q) = 1, e(y) = e 2πiy , and obtained some exact computational formulas for Actually the transforming formula (1) and are used, where p is an odd prime and 0 < h < p. Therefore, However, as far as we know, it seems that nobody has yet studied the hybrid mean value involving certain Hardy sums S 1 (h, k) and general Kloosterman sums K(r, l, λ; p), which are defined as follows: where λ is a Dirichlet character modulo q. If λ = λ 0 is the principal character modulo q and l = 1, then K(r, 1, λ 0 ; q) = K(r, q). Inspired by [15], we will study the hybrid mean value Since they are more comprehensive than those in [15], we turn to some novel methods and finally obtain several explicit formulas. Our results not only cover those of [15], but also contain something quite new.
Theorem 1. Let p be an odd prime. Then for any character λ mod p and any integers n, t with (n, p) = (t, p) = 1, if p ≡ 1 mod 4, we have the identity where χ is any odd character modulo p and χ 0 is the principal character modulo p.

Note:
The emergences of χ in Theorem 1 are due to the application of (2) in the proof.
Theorem 2. Let p be an odd prime. Then for any character λ mod p and any integers n, t with (n, p) = (t, p) = 1, if p ≡ 1 mod 4, we have the identity where h p denotes the class number of the quadratic field Q( √ −p), χ is any odd character modulo p, and τ(χ) = p−1 ∑ a=1 χ(a)e a p denotes the classical Gauss sums.

Corollary 2.
Let p be an odd prime, then we have It should be pointed out that we only consider the prime modulus case. The question of whether there exist some exact computational formulas for the general modulus q remains open.

Some Lemmas
In this section, we will give several simple lemmas, which are necessary in the proof of our theorems. Hereinafter, we shall use some knowledge of elementary number theory, the orthogonality relation of character sum and the properties of Gauss sums, which all can be found in [16], here we only list a few. For example, if (u, p) = 1 or χ is a primitive character modulo p. If χ is a primitive character modulo p, then |τ(χ)| = √ p.
Now we have the following lemmas: Lemma 1. Let p be an odd prime, n be any integer with (n, p) = 1. Then for any non-principal character χ mod p and any character λ mod p, we have Proof. From the properties of Gauss sums and the reduced residue system modulo p, we have Using the properties of Gauss sums and the periodicity of Dirichlet character, we have Applying the orthogonality relation for character modulo p, it is clear that And note that τ(χ 0 ) = −1, so we can get Note that |χ(−n)| = 1 and |τ(λχ)| = |τ(λχ)| = |τ(χ)| = √ p if λχ and λχ are all non-principal characters modulo p. Thus, from (3) and (4) we may immediately deduce that This proves Lemma 1.

Lemma 2.
Let p be an odd prime, n be any integer with (n, p) = 1. Then for any non-principal character χ mod p and any character λ mod p, we have Proof. From the definition of Kloosterman sums and the properties of Gauss sums we have Note that |τ(χ)| = √ p, |λχ(n)| = 1, τ(χ 0 ) = −1, then we can immediately deduce that This proves Lemma 2.
Lemma 3. Let p be an odd prime, then we have Proof. See Lemma 5 of [15].