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

Chang Leran; Li Xiaoxue

doi:10.1515/math-2016-0040

Enjoy 40% off

academic books on De Gruyter Brill *

Article Open Access

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

Chang Leran and Li Xiaoxue

Published/Copyright: June 27, 2016

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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.

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

S(h,q)=∑a=1qaqahq,

where

((x))=0,ifxisaninteger,x−[x]−12,ifxisnotaninteger;

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 [10] studied the hybrid mean value problem of Dedekind sums and Kloosterman sums

K(m,n;q)=∑∑′a=1qema+na¯q,

where q ≥ 3 is an integer, ∑∑′a=1q denotes the summation over all 1 < a <q with (a, q) = 1, e(y) = e^2πiy, 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 identity

∑a=1p−1∑b=1p−1K(a,1;p)2⋅K(b,1;p)2⋅S2(a⋅b¯,p)=112⋅p2⋅(p−1)(p−2),ifp≡1mod4,112⋅p2⋅((p−1)(p−2)−12⋅hp2),ifp≡3mod4;

where h_p denotes the class number of the quadratic field Q(−p).

Theorem B

Let p be an odd prime, then one has the asymptotic formula

∑a=1p−1∑b=1p−1K(a,1;p)2⋅K(b,1;p)2⋅S2(a⋅b¯,q)=124p5+0p4⋅exp3InInpInp,

where exp(y) = 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) = 1, one has the identity

∑a=1p−1∑n=0p−1en3+anp6=5p4−8p3−p2.

It is natural that one will ask, for the two-term exponential sums

E(r,s)=∑n=0p−1ern3+snp,

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 identity

∑a=1p−1∑b=1p−1E(a,1)2⋅E(b,1)2⋅S(a⋅b¯,p)=112(p−2)(p−1)p2−p2hp2,ifp≡3mod4,112(p−2)(p−1)p2ifp≡1mod4;

where h_p 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 formula

∑a=1p−1∑b=1p−1E(a,1)2⋅E(b,1)2⋅S2(a⋅b¯,p)=124⋅p5+0p4⋅exp3InInpInp.

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 identity

∑n=0p−1Xan2+bn+c=X¯(4)τ(X2)τ(X¯X2)τ(X¯)X4c−b2a¯4ac−b2p,

where X2=∗p denotes the Legendre symbol mod p.

Proof

From the definition and properties of Gauss sums we have

∑n=0p−1X(an2+bn+c)=1τ(X¯)∑r=1p−1X¯(r)∑n=0p−1er(an2+bn+c)p=X¯(4)τ(X¯)∑r=1p−1X¯(r)e4rc−rb2a¯)p∑n=0p−1era(2n+ba¯)2)p=X¯(4)τ(X¯)∑r=1p−1X¯(r)er(4c−rb2a¯)p∑n=0p−1eran2)p.(1)

For any integer a with (a, p) = 1, from Theorem 7.5.4 of [14] we know that

∑n=0p−1ean2p=ap⋅τ(X2).(2)

Combining (1) and (2) we have the identity

∑n=0p−1Xan2+bn+c=X¯(4)τ(X2)τ(X¯)ap∑r=1p−1X¯(r)rper(4c−b2a¯)p=X¯(4)τ(X2)τ(X¯X2)τ(X¯)X(4c−b2a¯)4ac−b2p.

This proves Lemma 2.1. □

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

∑a=1p−1X(a)∑n=0p−1ean3+np2=X(4)X2(3)τ(X2)τ(XX2)τ(X¯3),

Proof

From Lemma 2.1 the definition and properties of Gauss sums we have

∑a=1p−1X(a)∑n=0p−1ean3+np2=∑m=0p−1∑n=0p−1∑a=1p−1X(a)ea(m3−n3)+m−np=τ(X)∑m=0p−1∑n=0p−1X¯m3−n3em−np=τ(X)∑m=0p−1X¯3(m)emp+τ(X)∑m=0p−1∑n=1p−1X¯m3−n3em−np=τ(X)∑m=0p−1X¯3(m)emp+τ(x)∑m=0p−1X¯m3−1∑n=1p−1X¯3m3−1∑n=1p−1X¯3(n)en(m−1)p=τXτX¯3+τXτX¯3∑m=0p−1Xm−13X¯m3−1=2τXτX¯3+τXτX¯3∑m=2p−1Xm−13X¯m3−1=2τXτX¯3+τXτX¯3∑m=1p−2Xm3X¯m3+3m2+3m=2τXτX¯3+τXτX¯3∑m=1p−2X¯1+3m¯2+3m¯=2τXτX¯3+τXτX¯3∑m=1p−2X¯3m2+3m+1=τXτX¯3∑m=0p−1X¯3m2+3m+1=X(4)X2(3)τ(X2)τ(XX2)τ(X¯3),

This proves Lemma 2.2 □

Lemma 2.3

Let q > 2 be an integer. Then for any integer a with (a, q) = 1, we have the identity

S(a,q)=1π2q∑d/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

Proof

See Lemma 2 of [9]. □

Lemma 2.4

For any odd prime p, we have the asymptotic formula

∑XmoddX(−1)=−1L(1,X)4=5144⋅π4⋅p+oexp3InInpInp.

Proof

See Lemma 6 of [15]. □

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

S(a,p)=1π2⋅pp−1⋅∑XmoddX(−1)=−1X(a)L(1,X)2(3)

and

∑XmoddX(−1)=−1L(1,X)2=π2(p−1)p∑a=1p−1ap−122=π212⋅(p−1)2(p−2)p2.(4)

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

∑a=1p−1∑b=1p−1E(a,1)2⋅E(b,1)2⋅S(a⋅b¯,p)=1π2⋅pp−1⋅∑XmoddX(−1)=−1X(4)X2(3)τ(X2)τ(XX2)τ(X¯3)2⋅L(1,X)2=1π2⋅p3p−1⋅∑XmoddX(−1)=−1τ(XX2)2⋅L(1,X)2.(5)

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

∑a=1p−1∑b=1p−1E(a,1)2⋅E(b,1)2⋅S(a⋅b¯,p)=1π2⋅p4p−1⋅∑XmoddX(−1)=−1L(1,X)2=(p−2)(p−1)p212.(6)

If p ≡ 1 mod 4, thenχ₂(–1) = — 1. This time we have |τ(χ2χ₂)| = 1 and τ(XX2)=p,X≠X2. Combining (5) and (6) we obtain

∑a=1p−1∑b=1p−1E(a,1)2⋅E(b,1)2⋅S(a⋅b¯,p)=1π2⋅p4p−1⋅∑XmoddX(−1)=−1L(1,X)2+1π2⋅p3p−1L(1,X)2−1π2⋅p4p−1L(1,X)2=(p−2)(p−1)p212−p3π2L(1,X)2=(p−2)(p−1)p212−p2⋅hp2,(7)

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

∑a=1p−1∑b=1p−1E(a,1)2⋅E(b,1)2⋅S(a⋅b¯,p)=1π4⋅p4(p−1)2⋅∑XmoddX(−1)=−1∑ηmodpη(−1)=−1∑a=1p−1X(a)η(a)E(a,1)2×∑b=1p−1X¯(b)η¯(b)E(b,1)2L(1,X)2⋅L(1,η)2=1π4p2(p−1)2∑XmodpX(−1)=−1∑ηmodpη(−1)=−1Xη≠Xo∑a=1p−1X(a)η(a)E(a,1)22L(1,X)2⋅L(1,η)2+1π4⋅p2(p−1)2⋅∑a=1p−1E(a,1)22∑XmodpX(−1)=−1L(1,X)4≡R1+R2.(8)

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

∑a=1p−1E(a,1)2=p2,

from Lemma 2.4 we have the asymptotic formula

R2=1π4⋅p6(p−1)25π4144⋅p+Oexp3InInpInp=5144⋅p5+Op4⋅exp3InInpInp.(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

∑XmodpX(−1)=−1∑ηmodpη(−1)=−1XηX2=XoL(1,X)2⋅L(1,η)2=∑XmodpX(−1)=−1L(1,X)2⋅L(1,XX2)2=O(p).

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

R1=1π4⋅p5(p−1)2∑XmodpX(−1)=−1∑ηmodpη(−1)=−1Xη≠XoL(1,X)2⋅L(1,η)2+1π4⋅p4(p−1)2∑XmodpX(−1)=−1∑ηmodpη(−1)=−1Xη=X2L(1,X)2⋅(L(1,η)2−1π5⋅p5(p−1)2∑XmodpX(−1)=−1∑ηmodpη(−1)=−1Xη=X2L(1,X)2⋅L(1,η)2=1π4⋅p5(p−1)2∑XmodpX(−1)=−1L(1,X)22+O(p4)=1π4⋅p5(p−1)2⋅π4144⋅(p−1)4(p−2)2p4+O(p4)1144⋅p5+O(p4).(10)

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

∑a=1p−1∑b=1p−1E(a,1)2⋅E(b,1)2⋅S2(a⋅b¯,p)=1144⋅p5+5144⋅p5+Op4⋅exp3InInpInp=124⋅p5+Op4⋅exp3InInpInp.

This completes the proofs of our results.

Competing interests

The authors declare that they have no competing interests.

Acknowledgement

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.

References

[1] Rademacher H., On the transformation of log ŋ(τ), J. Indian Math. Soc., 1955, 19, 25-30.Search in Google Scholar

[2] Rademacher H., E. Grosswald, Dedekind Sums, Carus Mathematical Monographs, Math. Assoc. Amer., Washington D.C., 1972.10.5948/UPO9781614440161Search in Google Scholar

[3] Apostol T. M., Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976.10.1007/978-1-4684-9910-0Search in Google Scholar

[4] Carlitz L., The reciprocity theorem for Dedekind sums, Pacific J. Math., 1953, 3, 523-527.10.2140/pjm.1953.3.523Search in Google Scholar

[5] Conrey J. B., Eric Fransen, Robert Klein and Clayton Scott, Mean values of Dedekind sums, J. of Number Theory, 1996, 56, 214-226.10.1006/jnth.1996.0014Search in Google Scholar

[6] Jia C. H., On the mean value of Dedekind sums, J. of Number Theory, 2001, 87, 173-188.10.1006/jnth.2000.2580Search in Google Scholar

[7] Mordell L. J., The reciprocity formula for Dedekind sums, Amer. J. Math., 1951, 73, 593-598.10.2307/2372310Search in Google Scholar

[8] Zhang W. P., A note on the mean square value of the Dedekind sums, Acta Math. Hung., 2000, 86, 275-289.10.1023/A:1006724724840Search in Google Scholar

[9] Zhang W. P., On the mean values of Dedekind Sums, J. Théor. Nombr. Bordx., 1996, 8, 429-442.10.5802/jtnb.179Search in Google Scholar

[10] Zhang W. P., Liu Y. N., A hybrid mean value related to the Dedekind sums and Kloosterman sums, Sci. China Math., 2010, 53, 2543-2550.10.1007/s11425-010-3153-1Search in Google Scholar

[11] Zhang W. P., Han D., On the sixth power mean of the two-term exponential sums, J. of Number Theory, 2014, 136, 403-413.10.1016/j.jnt.2013.10.022Search in Google Scholar

[12] Duke W., Iwaniec H., A relation between cubic exponential and Kloosterman sums, Contemporary Mathematics, 1993, 143, 255-258.10.1090/conm/143/00999Search in Google Scholar

[13] Apostol T. M., Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.10.1007/978-1-4757-5579-4Search in Google Scholar

[14] Hua L. K., Introduction to Number Theory, Science Press, Beijing, 1979.Search in Google Scholar

[15] Zhang W. P., Yi Y., He X. L., On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, J. of Number Theory, 2000, 84, 199-213.10.1006/jnth.2000.2515Search in Google Scholar

Received: 2016-5-12

Accepted: 2016-6-13

Published Online: 2016-6-27

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.