TSWJ The Scientific World Journal 1537-744X 2356-6140 Hindawi Publishing Corporation 10.1155/2014/726053 726053 Research Article On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums Chen Guohui 1 Zhang Han 2 Das Kinkar Ch 1 College of Mathematics and Statistics Hainan Normal University Hainan 571158 China hainnu.edu.cn 2 School of Mathematics Northwest University Xi’an, Shaanxi 710127 China nwu.edu.cn 2014 1672014 2014 01 04 2014 08 07 2014 16 7 2014 2014 Copyright © 2014 Guohui Chen and Han 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.

For any fixed integer k2 and integer r with r, p=1, it is clear that there exist k integers 1aip-1i=1, 2, , k such that a1a2akrmodp. Let N(k,r;p) denote the number of all a1, a2, ak such that a1a2akrmodp and 2†a1+a2+⋯ + ak. In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties of N(k,r;p) and give two interesting asymptotic formulae for it.

1. Introduction

Let p be an odd number. For each integer a with 1ap-1, it is clear that there exists one and only one a¯ with 0a¯p-1 such that a·a¯1modp. Let N(p) denote the number of all 1ap-1 in which a and a¯ are of opposite parity. Professor D. H. Lehmer  asked us to study N(p) or at least to say something nontrivial about it. It is known that N(p)2    or  0mod4 when p±1mod4. Some works related to the Lehmer problem can be found in references . For example, Zhang [2, 4] proved the asymptotic formula (1)N(p)=12·p+O(p1/2·ln2p). In this paper, we will study a new summation related to the Lehmer problem. For any fixed integer k2 and integer r with (r,p)=1, we define the sums N(k,r;p) as follows: (2)N(k,r;p)=12a1=1p-1a2=1p-1ak=1p-1a1a2akrmodp(1-(-1)a1+a2++ak). In fact, N(k,r;p) is a generalization of the Lehmer problem. For example, if k=2 and r=1, then from the definition of N(2,1;p) we have (3)N(2,1;p)=12a=1p-1b=1p-1ab1modp(1-(-1)a+b)=12a=1p-1(1-(-1)a+a¯)=N(p). So N(2,1;p) becomes N(p), the Lehmer problem.

Now we are concerned about the arithmetical properties of N(k,r;p). This problem is interesting, because it is a generalization of the Lehmer problem.

In this paper, we use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties of N(k,r;p) and give two interesting asymptotic formulae for it. That is, we will prove the following.

Theorem 1.

Let p be an odd prime. Then for any fixed integer k2 and integer r with (r,p)=1, we have the asymptotic formula (4)N(k,r;p)=12·(p-1)k-1+O(p(k-1)/2·lnkp).

In order to facilitate the description of Theorem 2, we need to give the definition of high-dimension Kloosterman sums C(k,m;q). Let q3 be an integer. For any integer m, we define (5)C(k,m;q)=a1=1q(a1,q)=1a2=1q(a2,q)=1ak=1q(ak,q)=1e(a1+a2++ak+ma¯1a¯2a¯kq), where e(y)=e2πiy,ai·a¯i1modp,i=1,2,,k.

About some arithmetical properties of C(k,m;q), one can find them in . Let E(k,r;p)=N(k,r;p)-(1/2)(p-1)k-1 denote the error term in the asymptotic formula of N(k,r;p). As another main content of this paper, we will study the asymptotic properties of the hybrid mean value of C(k-1,r2¯k;p) and E(k,r;p) and also give a sharp asymptotic formula for it. That is, we will prove the following.

Theorem 2.

Let p be an odd prime. Then for any fixed integer k2, we have the asymptotic formula (6)r=1p-1C(k-1,r2¯k;p)·E(k,r;p)=-4k-1·ikπk·pk+O(pk-1+ϵ), where i2=-1, ϵ denotes any fixed positive number.

The constants 2¯k in Theorem 2 cannot be omitted. Otherwise, the main term in Theorem 2 is zero. If k=3 and 4, then from Theorem 2 we can also deduce the following two corollaries.

Corollary 3.

Let p be an odd prime. Then for any fixed positive number ϵ>0, we have the asymptotic formula (7)r=1p-1C(2,8¯r;p)·E(3,r;p)=16·iπ3·p3+O(p2+ϵ).

Corollary 4.

Let p be an odd prime. Then for any fixed positive number ϵ>0, we have the asymptotic formula (8)r=1p-1C(3,16¯r;p)·E(4,r;p)=-64π4·p4+O(p3+ϵ).

2. Several Lemmas

In this section, we will give several lemmas, which are necessary in the proofs of our theorems. Hereinafter, we will use many properties of Gauss sums and the estimate for high-dimension Kloosterman sums; all of these contents can be found in references [6, 9], so they will not be repeated here. First we have the following.

Lemma 5.

Let p be an odd prime. Then for fixed integer k1 and any integer m, we have the estimate (9)|C(k,m;p)|=|a1=1p-1a2=1p-1ak=1p-1e(a1+a2++ak+ma¯1a¯2a¯kp)|pk/2.

Proof.

See [6, 7].

Lemma 6.

Let p be an odd prime. Then for any odd character χmodp (i.e., χ(-1)=-1), we have the identity (10)a=1p-1(-1)a·χ(a)  =2(1-2χ(2))·(1p·a=1p-1a·χ(a)).

Proof.

See  or Lemma  3 in .

Lemma 7.

Let p be an odd prime. Then for any integer (r,p)=1, we have (11)N(k,r;p)=12·(p-1)k-1-2k-1·ikπk(p-1)×χ(-1)=-1χ¯(r)(1-2χ(2))kτk(χ)Lk(1,χ¯), where χ(-1)=-1 denotes the summation over all odd characters χmodp, τ(χ)=a=1p-1χ(a)·e(a/p) denotes the classical Gauss sums, and L(s,χ) denotes the Dirichlet L-function corresponding to χmodp.

Proof.

From the orthogonality of characters modp and the definition of N(k,r;p) we have the identity (12)N(k,r;p)=12a1=1p-1a2=1p-1ak=1p-1a1a2akrmodp(1-(-1)a1+a2++ak)=12·(p-1)k-1-12a1=1p-1a2=1p-1ak=1p-1a1a2akrmodp(-1)a1+a2++ak=12·(p-1)k-1-12(p-1)×χmodpχ¯(r)(a=1p-1(-1)a·χ(a))k12·(p-1)k-1-A(k,p). For any odd character χmodp, from Theorems 12.11 and 12.20 of  we have (13)1p·a=1p-1a·χ(a)=iπ·τ(χ)·L(1,χ¯). Note that, for any even character χmodp, we have the identity (14)a=1p-1(-1)a·χ(a)=0, from (13) and Lemma 6 we have (15)A(k,p)=2k-1·ikπk(p-1)×χ(-1)=-1χ¯(r)(1-2χ(2))kτk(χ)Lk(1,χ¯). Now Lemma 7 follows from (12) and (15).

Lemma 8.

Let p be an odd prime and k a fixed integer with k2. Then for any nonprincipal character χmodp and any real numbers yp3, we have the estimate (16)|nyχ(n)·dk(n)|y1-(1/2k-1)·p·lnp.

Proof.

We use mathematical induction to prove this lemma. If k=2, then from the Pòlya-Vinogradov inequality we have (17)|nyχ(n)d(n)|=|mnyχ(mn)|=|2nyχ(n)my/nχ(m)-(nyχ(n))2|yplnp=y1-(1/2)·p·lnp. Assume that the lemma holds for k=r. That is, (18)|nyχ(n)·dr(n)|y1-(1/2r-1)·p·lnp. Then for k=r+1, note that dr+1(n)=sndr(s); applying estimate (18) and the Pòlya-Vinogradov inequality we have (19)|nyχ(n)dr+1(n)|=|mnyχ(mn)dr(m)||nyχ(n)my/nχ(m)dr(m)+myχ(m)dr(m)ny/nχ(n)|+|nyχ(n)|·|myχ(m)dr(m)|ny(yn)1-(1/2r-1)·p·lnp+y1-(1/2r)·p·lnpy1-(1/2r)·p·lnp. Now our lemma follows from the induction.

3. Proofs of the Theorems

In this section, we will prove our conclusions. First we prove Theorem 1. For any real number Npk, applying Abel’s identity (see Theorem 4.2 of ) we have (20)Lk(1,χ¯)=1nNχ¯(n)dk(n)n+N1y2(N<nyχ¯(n)dk(n))dy. For any integer 0ik, from Lemma 5 and the definition of C(k,m;p) we have(21)χ(-1)=-1χ¯(r)χ(2i)τk(χ)1nNχ¯(n)dk(n)n=p-121nN(n,p)=1dk(n)n×a1=1p-1ak-1=1p-1e(a1++ak-1+nr2i¯·a¯1a¯2a¯k-1p)-p-121nN(n,p)=1dk(n)n×a1=1p-1ak-1=1p-1e(a1++ak-1-nr2i¯·a¯1a¯2a¯k-1p)p-121nN(n,p)=1dk(n)n·p(k-1)/2p(k+1)/2·lnkN.Applying (20) and the binomial expression we have the estimate (22)χ(-1)=-1χ¯(r)(1-2χ(2))kτk(χ)1nNχ¯(n)dk(n)np(k+1)/2·lnkN. Taking N=p2k-1, note that |τ(χ)|=p and the identity (23)N<nyχ¯(n)dk(n)=nyχ¯(n)dk(n)-nNχ¯(n)dk(n), and applying Lemma 8 we have the estimate (24)|χ(-1)=-1χ¯(r)(1-2χ(2))kτk(χ)nn×N1y2(N<nyχ¯(n)dk(n))dy|p(k+2)/2·pN1/2k-1·lnp=p(k+1)/2·lnp. Combining (20), (22), (24), and Lemma 7 we may immediately deduce the asymptotic formula (25)N(k,r;p)=12·(p-1)k-1+O(p(k-1)/2·lnkp). The proof of Theorem 1 is right.

Now we prove Theorem 2. For any nonprincipal character χmodp, from the definition and properties of Gauss sums we have(26)r=1p-1χ¯(r)·C(k-1,r2¯k;p)=r=1p-1χ¯(r)×a1=1p-1a2=1p-1ak-1=1p-1e(a1+a2++ak-1+r2¯ka¯1a¯2a¯k-1p)=a1=1p-1ak-1=1p-1e(a1+a2++ak-1p)×r=1p-1χ¯(r)e(r2¯ka¯1a¯2a¯k-1p)=τ(χ¯)χ¯(2k)×a1=1p-1ak-1=1p-1χ(a¯1a¯2a¯k-1)e(a1+a2++ak-1p)=χ¯(2k)·τk(χ¯).Note that τk(χ¯)·τk(χ)=χ¯k(-1)τ(χ)¯k·τk(χ)=χ¯k(-1)·pk; from Lemma 7 and the definition of E(k,r;p) we have (27)r=1p-1C(k-1,r2¯k;p)·E(k,r;p)=-2k-1·ikπk(p-1)·pk·χ(-1)=-1(-1)k(1-2χ(2))kχ¯(2k)Lk(1,χ¯)=(-1)k+12k-1·ikπk(p-1)·pk·j=0k(jk)(-1)j·2j·χ(-1)=-1χ¯(2k-j)n=1χ¯(n)·dk(n)n=(-1)k+12k-1·ikπk(p-1)·pk·p-12(-1)k·2k+O(pk-1+ϵ)=-4k-1·ikπk·pk+O(pk-1+ϵ), where i2=-1, (jk)=k!/(j!·(k-j)!), ϵ denotes any fixed positive number and dk(n) denotes the kth divisor function. That is, dk(n)=(d1nd2ndkn)d1d2dk=n1.

The proof of Theorem 2 is right.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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 the P. S. F. (2013JZ001) and N. S. F. (11371291) of China.

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