Let q be a natural number and h an integer prime to q. The classical Dedekind sums
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 [3–10]. For example, W. P. Zhang and Y. N. Liu  studied the hybrid mean value problem of Dedekind sums and Kloosterman sums
where q ≥ 3 is an integer, denotes the summation over all 1 < a < q with (a, q) = 1, e(y) = e2πiy, and ā 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 identitywhere hp denotes the class number of the quadratic field
Theorem B: Let p be an odd prime, then one has the asymptotic formulawhere exp(y) = ey.
On the other hand, W. P. Zhang and D. Han  studied the sixth power mean of the two-term exponential sums, and proved that for any prime p > 3 with (3, p — 1) = 1, one has the identity
It is natural that one will ask, for the two-term exponential sums
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) = 1, then we have the identitywhere hp denotes the class number of the quadratic field
Theorem 1.2: Let p > 3 be a prime with (3, p — 1) = 1, then we have the asymptotic formula
It is very interesting that the results in our paper are exactly the same as in reference . 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 .
2 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 , so they will not be repeated here.
Lemma 2.1: Let p be an odd prime, a be any integer with (a, p) = 1. For any non-principal character χ mod p, we have the identitywhere denotes the Legendre symbol mod p.
Lemma 2.2: Let p be an odd prime. Then for any non-principal character χ mod p with χ3 ≠ χ0 (the principal character mod p), we have the identity
Lemma 2.3: Let q > 2 be an integer. Then for any integer a with (a, q) = 1, we have the identitywhere L(1,χ) denotes the Dirichlet L-function corresponding to character χ mod d
Lemma 2.4: For any odd prime p, we have the asymptotic formula
3 Proof of the theorems
where we have used the identity(8)
Now we estimate R1 and R2 in (8) respectively. Note that the identity
from Lemma 2.4 we have the asymptotic formula(9)
If p ≡ 1 mod 4, then there exist two odd characters χ and ŋ such that χŋχ2 =χ0. This time, we have the estimate(10)
This completes the proofs of our results.
The authors declare that they have no competing interests.
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
This work is supported by N. S. F. (11371291) of P. R. China and G.I.C.F. (YZZ15009) of Northwest University.
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Published Online: 2016-06-27
Published in Print: 2016-01-01
© 2016 Leran and Xiaoxue, published by De Gruyter Open.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)