The hybrid mean value of Dedekind sums and two-term exponential sums

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


Introduction
Let q be a natural number and h an integer prime to q. The classical Dedekind sums where ..x// D ( x OEx 1 2 ; if x is not an integer; 0; if x is an integer, describes the behaviour of the logarithm of the eta-function (see [1,2]) under modular transformations. The various arithmetical properties of S.h; q/ were investigated by many authors, who obtained a series of results, see [3][4][5][6][7][8][9][10]. For example, W. P. Zhang and Y. N. Liu [10] studied the hybrid mean value problem of Dedekind sums and Kloosterman sums where q 3 is an integer, q X 0 aD1 denotes the summation over all 1 Ä a Ä q with .a; q/ D 1, e.y/ D e 2 iy , and a denotes the multiplicative inverse of a mod q. They proved the following results: Theorem A. Let p be an odd prime, then one has the identity p 1 where exp.y/ D e y .
On the other hand, W. P. Zhang and D. Han [11] studied the sixth power mean of the two-term exponential sums, and proved that for any prime p > 3 with .3; p 1/ D 1, one has the identity It is natural that one will ask, for the two-term exponential sums whether there exists an identity (or asymptotic formula) similar to Theorem A (or Theorem B). The answer is yes.
The main purpose of this paper is to show this point. That is, we shall use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to prove the following similar conclusions: Theorem 1.1. Let p > 3 be an odd prime with .3; p 1/ D 1, then we have the identity where h p denotes the class number of the quadratic field Q p p .
Theorem 1.2. Let p > 3 be a prime with .3; p 1/ D 1, then we have the asymptotic formula It is very interesting that the results in our paper are exactly the same as in reference [10]. This means that there is close relationship between Kloosterman sums and two-term exponential sums. In fact, some close relationships can be found in W. Duke and H. Iwaniec [12].

Several lemmas
To complete the proof of our theorems, we need to prove several lemmas. Hereinafter, we shall use some properties of characters mod q and Dirichlet L-functions, all of these can be found in reference [13], so they will not be repeated here.
Lemma 2.1. Let p be an odd prime, a be any integer with .a; p/ D 1. For any non-principal character mod p, we have the identity where 2 D p Á denotes the Legendre symbol mod p.
Proof. From the definition and properties of Gauss sums we have For any integer a with .a; p/ D 1, from Theorem 7:5:4 of [14] we know that Combining (1) and (2) we have the identity This proves Lemma 2.1.

Proof of the theorems
In this section, we shall complete the proof of our theorems. First we prove Theorem 1.
If p Á 1 mod 4, then 2 ¤ 0 for all odd character mod p. This time, from (4) and (5) where we have used the identity L.1; 2 / D h p = p p.