Abstract and Applied Analysis
Volume 2013, Issue 1 760505
Research Article
Open Access

On the Hybrid Mean Value Involving Kloosterman Sums and Sums Analogous to Dedekind Sums

Di Han, 

Corresponding Author

Di Han

Department of Mathematics, Northwest University, Xi′an, Shaanxi 710127, China nwu.edu.cn

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Wenpeng Zhang, 

Wenpeng Zhang

Department of Mathematics, Northwest University, Xi′an, Shaanxi 710127, China nwu.edu.cn

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First published: 31 October 2013
https://doi.org/10.1155/2013/760505
Academic Editor: Claudia Timofte

Abstract

The main purpose of this paper is using the properties of Gauss sums and the mean value theorem of Dirichlet L-functions to study one kind of hybrid mean value problems involving Kloosterman sums and sums analogous to Dedekind sums and give two exact computational formulae for them.

1. Introduction

Let c be a natural number and let d be an integer prime to c. The classical Dedekind sums
()
where
()
describe the behaviour of the logarithm of the eta-function (see [1, 2]) under modular transformations. Gandhi [3] also introduced another sum analogous to Dedekind sums S(h, k) as follows:
()
where k denotes any positive even number and h denotes any integer with (h, k) = 1.

About the arithmetical properties of S2(h, k) and related sums, many authors had studied them and obtained a series of interesting results; see [1–9]. For example, the second author [7] proved the following conclusion.

Let k = 2βM be a positive integer with β ≥ 1 and (M, 2) = 1. Then we have the asymptotic formula
()
where denotes the summation over all integers 1 ≤ h ≤ k such that (h, k) = 1, denotes the product over all prime divisors of M such that pα∣M and pα+1†M, ϕ(k) is the Euler function, and exp (y) = ey.

The sum S2(h, k) is important, because it has close relations with the classical Dedekind sums S(h, k). But unfortunately, so far, we knew that all results of S2(h, k) are the properties of their own, or the relationships between S2(h, k) and S(h, k), and had nothing to do with the other arithmetic functions. If we can find some relations between S2(h, k) and other arithmetic function, that will be very useful for further study of the properties of S2(h, k).

On the other hand, we introduce the classical Kloosterman sums K(n, q), which are defined as follows. For any positive integer q > 1 and integer n,
()
where denotes the solution of the congruence x · b ≡ 1mod q and e(x) = e2πix.

Some elementary properties of K(n, q) can be found in [10, 11].

The main purpose of this paper is using the properties of the Gauss sums and the mean square value theorem of Dirichlet L-functions to study a hybrid mean value problem involving S2(h, k) and Kloosterman sums and give two exact computational formulae for them. That is, we will prove the following.

Theorem 1. Let p be an odd prime. Then one has the identity

()
where .

Theorem 2. Let p be an odd prime; then one has the identity

()
where hp denotes the class number of the quadratic field .

2. Several Lemmas

In this section, we will give several lemmas, which are necessary in the proof of our theorems. Hereinafter, we will use many properties of Gauss sums, all of which can be found in [12], so they will not be repeated here. First we have the following.

Lemma 3. Let p be an odd prime; then one has the identity

()

Proof. It is clear that if n pass through a complete residue system mod p, then 2n − 1 also pass through a complete residue system mod p. So for any nonprincipal character χ mod p, from the properties of Gauss sums τ(χ) (see Theorem 8.9 of [12])

()
we have the identity
()
This proves Lemma 3.

Lemma 4. Let q > 2 be an integer; then for any integer a with (a, q) = 1, one has the identity

()
where L(1, χ) denotes the Dirichlet L-function corresponding to character χmod d.

Proof. See Lemma 2 of [8].

Lemma 5. Let p be an odd prime. Then for any odd number h with (h, p) = 1, one has the identity

()
where satisfies the congruence .

Proof. Note that the divisors of 2p are 1, 2, p, and 2p. So from Lemma 4 and the definition of S2(h, 2p) and S(h, k) we have

()
()
where λ denotes the principal character mod   2.

From the Euler infinite product formula (see Theorem 11.6 of [12]) we have,

()
where ∏p  denotes the product over all primes p.

From Lemma 4 we also have the identity

()
Note that h is an odd number; combining (14), (15), and (16) we have the identity
()
This proves Lemma 5.

Lemma 6. Let p be an odd prime. Then one has the identities

  • (A)

    ()

  • (B)

    ()

Proof. From the definition of Dedekind sums we have

()

If p ≡ 1mod c, then, from (20) and noting that the reciprocity theorem of Dedekind sums (see [5]), we have the computational formula

()
Now taking c = 1 in (21), from (16) we may immediately deduce the identity
()
Taking c = 2 in (21), from (16) we can also deduce the identity
()
Now Lemma 6 follows from (22) and (23).

3. Proof of the Theorems

In this section, we will complete the proof of our theorems. First we prove Theorem 1. Note that if χ is a nonprincipal character mod p, then and
()
From (24) and Lemmas 4, 5, and 6 we have
()
This proves Theorem 1.
Now we prove Theorem 2. If p ≡ 1mod 4, then from Lemmas 3, 5, and 6 we have
()
If p ≡ 3mod 4, then note that the Legendre symbol (−1/p) = χ2(−1) = −1, (see Dirichlet′s class number formula, Chapter 6 of [13]), and
()
so from Lemmas 3, 5, and 6 we have
()
Note that if p ≡ 3mod 8; and (2/p) = 1 if p ≡ 7mod 8, from (28) we may immediately deduce
()
Now Theorem 2 follows from (26) and (29).

This completes the proofs of all results.

Acknowledgments

The authors would like to thank the referee for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the NSF (11071194) of China.

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