AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 760505 10.1155/2013/760505 760505 Research Article On the Hybrid Mean Value Involving Kloosterman Sums and Sums Analogous to Dedekind Sums http://orcid.org/0000-0002-0335-3579 Han Di Zhang Wenpeng Timofte Claudia Department of Mathematics Northwest University Xi'an Shaanxi 710127 China nwu.edu.cn 2013 31 10 2013 2013 03 07 2013 19 09 2013 2013 Copyright © 2013 Di Han and Wenpeng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 (1)S(d,c)=j=1c((jc))((djc)), where (2)((x))=  {x-[x]-12,if  x  is  not  an  integer;0,if  x  is  an  integer,describe the behaviour of the logarithm of the eta-function (see [1, 2]) under modular transformations. Gandhi  also introduced another sum analogous to Dedekind sums S(h,k) as follows: (3)S2(h,k)=j=1k(-1)j((jk))((hjk)), 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 . For example, the second author  proved the following conclusion.

Let k=2βM be a positive integer with β1 and (M,2)=1. Then we have the asymptotic formula (4)h=1k|S2(h,k)|2=5112kϕ(k)(35-222β)×pαM[(1+(1/p))2-(1/p3α+1)](1+(1/p)+(1/p2))+O(k·exp(4lnlnklnk)), where h=1k denotes the summation over all integers 1hk such that (h,k)=1, pαM denotes the product over all prime divisors of M such that pαM and pα+1M, 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, (5)K(n,q)=b=1qe(nb+b-q), where b¯ denotes the solution of the congruence x·b1modq 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 (6)m=1p(2m-1,p)=1n=1p(2n-1,p)=1K(2m-1,p)·K(2n-1,p)dddddddddddddd·S2((2m-1)·2n-1¯,2p)=-p(p-1)4, where (2n-1)·(2n-1)¯1mod2p.

Theorem 2.

Let p be an odd prime; then one has the identity (7)m=1p(2m-1,p)=1n=1p(2n-1,p)=1|K(2m-1,p)|2·|K(2n-1,p)|2ddddddddddddddd·S2((2m-1)·(2n-1)¯,p)={-14p2(p-1)+3·p2·hp2, if p3mod8;-14p2(p-1)-p2·hp2, if   p7mod8;-14p2(p-1), if p1mod4, where hp denotes the class number of the quadratic field Q(-p).

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 , 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 (8)n=1pχ(2n-1)·|K(2n-1,p)|2=χ¯(-1)·τ3(χ)·τ(χ¯2)τ(χ¯).

Proof.

It is clear that if n pass through a complete residue system modp, then 2n-1 also pass through a complete residue system modp. So for any nonprincipal character χmodp, from the properties of Gauss sums τ(χ) (see Theorem 8.9 of ) (9)χ(a)=1τ(χ¯)b=1p-1χ¯(b)e(bap), we have the identity (10)n=1pχ(2n-1)·|K(2n-1,p)|2=n=1pχ(n)|K(n,p)|2=a=1p-1b=1p-1n=1p-1χ(n)e(n(a-b)+(a¯-b¯)p)=a=1p-1b=1p-1n=1p-1χ(n)e(nb(a-1)+b¯(a¯-1)p)=τ(χ)·a=1p-1b=1p-1χ¯(b(a-1))e(b¯(a¯-1)p)=τ2(χ)·a=1p-1χ¯(a-1)χ¯(a¯-1)=τ2(χ)·a=1p-1χ(a)χ¯(-(a-1)2)=χ¯(-1)·τ2(χ)·a=1p-2χ(a+1)χ¯(a2)=χ¯(-1)·τ2(χ)·a=1p-2χ(a¯+a¯2)=χ¯(-1)·τ2(χ)·a=1p-1χ(a2+a)=χ¯(-1)·τ2(χ)·1τ(χ¯)a=1p-1b=1p-1χ¯(b)e(b(a2+a)p)=χ¯(-1)·τ2(χ)·1τ(χ¯)·b=1p-1χ¯(b)e(bp)a=1p-1χ(a)e(bap)=χ¯(-1)·τ3(χ)·1τ(χ¯)·b=1p-1χ¯2(b)e(bp)=χ¯(-1)·τ3(χ)·τ(χ¯2)τ(χ¯). 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 (11)S(a,q)=1π2qd|qd2ϕ(d)χmoddχ(-1)=-1χ(a)|L(1,χ)|2, where L(1,χ) denotes the Dirichlet L-function corresponding to character χmodd.

Proof.

See Lemma 2 of .

Lemma 5.

Let p be an odd prime. Then for any odd number h with (h,p)=1, one has the identity (12)S2(h,2p)=-S(h,p)+S(2h,p)+S(2¯h,p), where 2¯ satisfies the congruence 2·2¯1modp.

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 (13)S2(h,2p)=j=12p(-1)j((j2p))((hj2p))=2j=1p((jp))((hjp))-j=12p((j2p))((hj2p))=2S(h,p)-S(h,2p)=2S(h,p)-12π2pd|2pd2ϕ(d)χmoddχ(-1)=-1χ(h)|L(1,χ)|2=2S(h,p)-p2π2(p-1)χmodpχ(-1)=-1χ(h)|L(1,χ)|2(14)-2pπ2(p-1)χmod2pχ(-1)=-1χ(h)|L(1,χ)|2=2S(h,p)-p2π2(p-1)χmodpχ(-1)=-1χ(h)|L(1,χ)|2-2pπ2(p-1)χmodpχ(-1)=-1χ(h)λ(h)|L(1,χλ)|2, where λ denotes the principal character mod  2.

From the Euler infinite product formula (see Theorem 11.6 of ) we have, (15)|L(1,χλ)|2=p1|1-χ(p1)λ(p1)p1|-2=p1>2|1-χ(p1)p1|-2=|1-χ(2)2|2·p1|1-χ(p1)p1|-2=(54-χ(2)2-χ¯(2)2)·|L(1,χ)|2, where p denotes the product over all primes p.

From Lemma 4 we also have the identity (16)S(h,p)=1π2·pp-1χmodpχ(-1)=-1χ(h)|L(1,χ)|2. Note that h is an odd number; combining (14), (15), and (16) we have the identity (17)S2(h,2p)=2S(h,p)-12S(h,p)-2pπ2(p-1)·χmodpχ(-1)=-1χ(h)|L(1,χλ)|2=32S(h,p)-2pπ2(p-1)·χmodpχ(-1)=-1χ(h)(54-χ(2)2-χ¯(2)2)·|L(1,χ)|2=-S(h,p)+S(2h,p)+S(2¯h,p). This proves Lemma 5.

Lemma 6.

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

(18)χmodpχ(-1)=-1|L(1,χ)|2=π212·(p-1)2·(p-2)p2;

(19)χmodpχ(-1)=-1χ(2)·|L(1,χ)|2=π224·(p-1)2·(p-5)p2.

Proof.

From the definition of Dedekind sums we have (20)S(1,c)=a=1c-1(ac-12)2=(c-1)(c-2)12c.

If p1modc, then, from (20) and noting that the reciprocity theorem of Dedekind sums (see ), we have the computational formula (21)S(c,p)=p2+c2+112pc-14-S(p,c)=p2+c2+112pc-14-S(1,c)=p2+c2+112pc-14-(c-1)(c-2)12c=(p-1)(p-1-c2)12pc. Now taking c=1 in (21), from (16) we may immediately deduce the identity (22)χmodpχ(-1)=-1|L(1,χ)|2=π212·(p-1)2·(p-2)p2. Taking c=2 in (21), from (16) we can also deduce the identity (23)χmodpχ(-1)=-1χ(2)·|L(1,χ)|2=π224·(p-1)2·(p-5)p2. 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 modp, then |τ(χ)|=p and (24)|m=1pχ(2m-1)K(2m-1,p)|=|a=1p-1m=1pχ(m)e(ma+a¯p)|=|τ2(χ)|=p. From (24) and Lemmas 4, 5, and 6 we have (25)m=1p(2m-1,p)=1n=1p(2n-1,p)=1K(2m-1,p)·K(2n-1,p)dddddddddddddd·S2((2m-1)·2n-1¯,2p)=-pπ2(p-1)χmodpχ(-1)=-1|n=1pχ(2n-1)·K(2n-1,p)|2dddddddddddddddsdds·|L(1,χ)|2+pπ2(p-1)×χmodpχ(-1)=-1χ(2)|n=1pχ(2n-1)·K(2n-1,p)|2dddddddddsdds·|L(1,χ)|2+pπ2(p-1)×χmodpχ(-1)=-1χ¯(2)|n=1pχ(2n-1)·K(2n-1,p)|2dddddddddsdds·|L(1,χ)|2=-p3π2(p-1)χmodpχ(-1)=-1|L(1,χ)|2+p3π2(p-1)×χmodpχ(-1)=-1χ(2)|L(1,χ)|2+p3π2(p-1)×χmodpχ(-1)=-1χ¯(2)·|L(1,χ)|2=-p(p-1)(p-2)12+p(p-1)(p-5)12=-p(p-1)4. This proves Theorem 1.

Now we prove Theorem 2. If p1mod4, then from Lemmas 3, 5, and 6 we have (26)m=1p(2m-1,p)=1n=1p(2n-1,p)=1|K(2m-1,p)|2·|K(2n-1,p)|2ddddddddddddddd·S2((2m-1)·(2n-1)¯,p)=-pπ2(p-1)χmodpχ(-1)=-1|n=1pχ(2n-1)·|K(2n-1,p)|2|2dddddddddddddddsdds·|L(1,χ)|2+pπ2(p-1)×χmodpχ(-1)=-1χ(2)|n=1pχ(2n-1)·|K(2n-1,p)|2|2dddddddddsdds·|L(1,χ)|2+pπ2(p-1)×χmodpχ(-1)=-1χ¯(2)|n=1pχ(2n-1)·|K(2n-1,p)|2|2dddsddddsdds·|L(1,χ)|2=-p4π2(p-1)χmodpχ(-1)=-1|L(1,χ)|2+p4π2(p-1)×χmodpχ(-1)=-1χ(2)|L(1,χ)|2+p4π2(p-1)×χmodpχ(-1)=-1χ¯(2)·|L(1,χ)|2=-112p2(p-1)(p-2)+112p2(p-1)(p-5)=-14p2(p-1). If p3mod4, then note that the Legendre symbol (-1/p)=χ2(-1)=-1, L(1,χ2)=π·hp/p (see Dirichlet's class number formula, Chapter 6 of ), and (27)τ(χ22)=a=1p-1(ap)2e(ap)=a=1p-1e(ap)=-1, so from Lemmas 3, 5, and 6 we have (28)m=1p(2m-1,p)=1n=1p(2n-1,p)=1|K(2m-1,p)|2·|K(2n-1,p)|2ddddddddddddddd·S2((2m-1)·(2n-1)¯,p)=-pπ2(p-1)χmodpχ(-1)=-1|n=1pχ(2n-1)·|K(2n-1,p)|2|2dddddddddddddddsdds·|L(1,χ)|2+pπ2(p-1)×χmodpχ(-1)=-1χ(2)|n=1pχ(2n-1)·|K(2n-1,p)|2|2dddddddddsdds·|L(1,χ)|2+pπ2(p-1)×χmodpχ(-1)=-1χ¯(2)|n=1pχ(2n-1)·|K(2n-1,p)|2|2dddddddddsdds·|L(1,χ)|2=-p4π2(p-1)χmodpχ(-1)=-1|L(1,χ)|2+p4π2(p-1)×χmodpχ(-1)=-1χ(2)|L(1,χ)|2+p4π2(p-1)×χmodpχ(-1)=-1χ¯(2)·|L(1,χ)|2+p3π2·|L(1,χ2)|2-p3π2·(2p)·|L(1,χ2)|2-p3π2·(2p)·|L(1,χ2)|2=-14p2(p-1)+p2·hp2-2(2p)·p2·hp2. Note that (2/p)=(-1)(p2-1)/8=-1 if p3mod8; and (2/p)=1 if p7mod8, from (28) we may immediately deduce (29)m=1p(2m-1,p)=1n=1p(2n-1,p)=1|K(2m-1,p)|2·|K(2n-1,p)|2ddddddddddddddd·S2((2m-1)·(2n-1)¯,p)={-14p2(p-1)+3·p2·hp2,ifp3mod8;-14p2(p-1)-p2·hp2,ifp7mod8. 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.

Rademacher H. On the transformation of log ητ Journal of the Indian Mathematical Society 1955 19 25 30 Rademacher H. Dedekind Sums, Carus Mathematical Monographs 1972 Washington, DC, USA Mathematical Association of America Gandhi J. M. On sums analogous to Dedekind sums Proceedings of the 5th Manitoba Conference on Numerical Mathematics 1975 Winnipeg, Mantiboba 647 655 Berndt B. C. Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan Journal für die Reine und Angewandte Mathematik 2009 1978 303-304 332 365 Carlitz L. The reciprocity theorem of Dedekind sums Pacific Journal of Mathematics 1953 3 3 513 522 10.2140/pjm.1953.3.513 Conrey J. B. Fransen E. Klein R. Scott C. Mean values of Dedekind sums Journal of Number Theory 1996 56 2 214 226 2-s2.0-0030079980 10.1006/jnth.1996.0014 ZBL0851.11028 Zhang W. A sum analogous to the Dedekind sum and its mean value formula Journal of Number Theory 2001 89 1 1 13 2-s2.0-0041738893 10.1006/jnth.2000.2624 ZBL0997.11076 Wenpeng Z. On the mean values of Dedekind sums Journal de Theorie des Nombres de BorDeaux 1996 8 2 429 442 10.5802/jtnb.179 Sitaramachandraro R. Dedekind and Hardy sums Acta Arithmetica 1987 48 4 325 340 Chowla S. On Kloosterman's sums Det Kongelige Norske Videnskabers Selskabs Forhandlinger 1967 40 70 72 Malyshev A. V. A generalization of Kloosterman sums and their estimates Vestnik Leningrad University 1960 15 59 75 Apostol T. M. Introduction to Analytic Number Theory 1976 New York, NY, USA Springer Davenport H. Multiplicative Number Theory 1967 Chicago, Ill, USA Markham