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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)


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

Chang Leran
  • School of Mathematics, Northwest University, Xi’an, Shaanxi, China
/ Li Xiaoxue
  • Corresponding author
  • School of Mathematics, Northwest University, Xi’an, Shaanxi, China
  • Email:
Published Online: 2016-06-27 | DOI: https://doi.org/10.1515/math-2016-0040


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.

Keywords: Dedekind sums; The two-term exponential sums; Hybrid mean value; Identity; Asymptotic formula

MSC 2010: 11L03; 11F20

1 Introduction

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 [310]. 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, a=1q 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 identitya=1p1b=1p1K(a,1;p)2K(b,1;p)2S2(ab¯,p)=112p2(p1)(p2),ifp1mod4,112p2((p1)(p2)12hp2),ifp3mod4;where hp denotes the class number of the quadratic field Q(p).

Theorem B: Let p be an odd prime, then one has the asymptotic formulaa=1p1b=1p1K(a,1;p)2K(b,1;p)2S2(ab¯,q)=124p5+0p4exp3InInpInp,where exp(y) = ey.

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) = 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) = 1, then we have the identitya=1p1b=1p1E(a,1)2E(b,1)2S(ab¯,p)=112(p2)(p1)p2p2hp2,ifp3mod4,112(p2)(p1)p2ifp1mod4;where hp denotes the class number of the quadratic field Q(p).

Theorem 1.2: Let p > 3 be a prime with (3, p — 1) = 1, then we have the asymptotic formulaa=1p1b=1p1E(a,1)2E(b,1)2S2(ab¯,p)=124p5+0p4exp3InInpInp.

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].

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 [13], 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 identityn=0p1Xan2+bn+c=X¯(4)τ(X2)τ(X¯X2)τ(X¯)X4cb2a¯4acb2p,where X2=p 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 identitya=1p1X(a)n=0p1ean3+np2=X(4)X2(3)τ(X2)τ(XX2)τ(X¯3),

Lemma 2.3: Let q > 2 be an integer. Then for any integer a with (a, q) = 1, we have the identityS(a,q)=1π2qd/qd2ϕ(d)XmoddX(1)=1X(a)L(1,X)2,where 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 formulaXmoddX(1)=1L(1,X)4=5144π4p+oexp3InInpInp.

3 Proof of the theorems

In this section, we shall complete the proof of our theorems. First we prove Theorem 1.1. From Lemma 2.3. with q = p (an odd prime) we have




It is clear that if (3, p — 1) = 1, then for any odd character χ mod p, we have χ3 ≠ χ0, the principal character mod p. Note that τ(X)=p, if χ ≠χ0. So from (3) and Lemma 2.2. we have


If p ≡ 1 mod 4, then χχ2 ≠ χ0 for all odd character χ mod p. This time, from (4) and (5) we have


If p ≡ 1 mod 4, thenχ2(–1) = — 1. This time we have |τ(χ2χ2)| = 1 and τ(XX2)=p,XX2. Combining (5) and (6) we obtain


where we have used the identity L(1,X2)=πhp/p.

Combining (6) and (7) we may immediately deduce Theorem 1.1.

Now we prove Theorem 1.2. From (3) we have


Now we estimate R1 and R2 in (8) respectively. Note that the identity


from Lemma 2.4 we have the asymptotic formula


If p ≡ 1 mod 4, then there exist two odd characters χ and ŋ such that χŋχ2 =χ0. This time, we have the estimate


So for any prime p > 3, from (4), Lemma 2.2 and Lemma 2.4 we also have the asymptotic formula


Combining (8), (9) and (10) we have the asymptotic formula


This completes the proofs of our results.

Competing interests

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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About the article

Received: 2016-05-12

Accepted: 2016-06-13

Published Online: 2016-06-27

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0040. Export Citation

© 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)

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