New identities involving Hardy sums S3(h, k) and general Kloosterman sums

Abstract: The main purpose of this paper is to obtain some exact computational formulas or upper bounds for hybrid mean value involving Hardy sums S3(h, p) and general Kloosterman sums K(r, l, λ; p). By applying the properties of Gauss sums and the mean value theorems of Dirichlet L-function, we derive some new identities. As the special cases, we also deduce some exact computational formulas for hybrid mean value involving S3(h, p) and classical Kloosterman sums K(n, p).


Introduction and main results
If h and k are integers with k > 0, the classical Dedekind sums S (h, k) are defined as if x is an integer.
The various properties of S (h, k) were investigated by many authors, one of which is reciprocity theorem (see Tom M. Apostol [1] or L. Carlitz [2]). That is, for all positive integers h and k with (h, k) = 1, we have the identity Conrey et al. [3] studied the mean value distribution of S (h, k) and deduced the important asymptotic formula Moreover, X. L. He and W. P. Zhang [4] gave an interesting asymptotic formula for the Dedekind sums with a weight of Hurwitz zeta-function as follows: Other sums analogous to the Dedekind sums are the Hardy sums. Using the notation of Berndt and Goldberg [5], they defined where h and k are integers with k > 0.
In 2014, H. Zhang and W. P. Zhang [6] obtained some beautiful identities involving S 1 (h, k) in the forms of where K(n, p) denotes the reduced form of the general Kloosterman sums attached to a Dirichlet character λ modulo k as where e(x) = e 2πix , a denotes the solution of the congruence x · a ≡ 1 mod k.
Recently, H. F. Zhang and T. P. Zhang [7] extended the results in [6] to a more general situation as K(m, s, λ; p)K(n, t, λ; p)S 1 (2mn, p), where K(n, t, λ; p) denotes complex conjugate of K(n, t, λ; p). Actually there are six forms of Hardy sums (see Berndt [8] and Goldberg [9]). A natural question is whether we can obtain similar results by replacing S 1 (h, k) with other forms of Hardy sums. Due to some technical reasons, for most of other forms of Hardy sums, the answer is no! Thanks to the important relationships among Hardy sums and Dedekind sums built by R. Sitaramachandrarao [10], we are lucky to find the only one S 3 (h, p) to replace, with Our starting point relies heavily on the following in [10] as: Proposition 1. Let k be an odd positive integer, h be an integer with (h, k) = 1. Then Then applying the properties of Gauss sums and the mean square value of Dirichlet L-functions, we have Theorem 1. Let p be an odd prime. Then for any Dirichlet character λ mod p and any integer s, t with (s, p) = (t, p) = 1, we have where χ is an odd Dirichlet character modulo p and χ 0 is the principal character modulo p. Theorem 2. Let p be an odd prime with p ≡ 1 mod 4. Then for any Dirichlet character λ mod p and any integer s, t with (s, p) = (t, p) = 1, we have Theorem 3. Let p be an odd prime with p ≡ 3 mod 8. Then for any Dirichlet character λ mod p and any integer s, t with (s, p) = (t, p) = 1, we have where h p denotes the class number of the quadratic field Q √ −p . Theorem 4. Let p be an odd prime with p ≡ 7 mod 8. Then for any Dirichlet character λ mod p and any integer s, t with (s, p) = (t, p) = 1, we have Taking λ = λ 0 , s = t = 1 in Theorems 1-4, we immediately deduce the following results. Corollary 1. Let p be an odd prime. Then we have the identity Corollary 2. Let p be an odd prime. Then we have

Some Lemmas
To prove the Theorems, we need the following Lemmas. Lemma 1. Let k > 2 be an integer. Then for any integer a with (a, k) = 1, we have the identity where L(1, χ) denotes the Dirichlet L-function corresponding to Dirichlet character χ mod d.
Proof. See Lemma 2 of [11]. Lemma 2. Let p be an odd prime, s be any integer with (s, p) = 1. Then for any non-principal character χ mod p and any Dirichlet character λ mod p, we have Proof. See Lemma 2 of reference [7]. Lemma 3. Let p be an odd prime, s be any integer with (s, p) = 1. Then for any non-principal character χ mod p and any Dirichlet character λ mod p, we have if λχ χ 0 , λχ χ 0 ; where τ(χ) = p a=1 χ(a)e a p denotes the classical Gauss sums.
Proof. See Lemma 1 of reference [7]. Lemma 4. Let p be an odd prime, then we have Proof. See Lemma 5 of reference [6].