On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

Abstract

For any fixed integer 𝑘≥2 and integer 𝑟 with (𝑟,𝑝)=1, it is clear that there exist k integers 1≤𝑎_𝑖≤𝑝−1 (𝑖=1,2,…,𝑘) such that 𝑎₁𝑎₂⋯𝑎_𝑘≡𝑟mod𝑝. Let 𝑁(𝑘,𝑟;𝑝) denote the number of all (𝑎₁,𝑎₂,⋯𝑎_𝑘) such that 𝑎₁𝑎₂⋯𝑎_𝑘≡𝑟mod𝑝 and 2†(𝑎₁+𝑎₂+⋯+𝑎_𝑘). 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 1≤𝑎≤𝑝−1, it is clear that there exists one and only one 𝑎 with 0≤𝑎≤𝑝−1 such that 𝑎⋅𝑎≡1mod𝑝. Let 𝑁(𝑝) denote the number of all 1≤𝑎≤𝑝−1 in which 𝑎 and 𝑎 are of opposite parity. Professor D. H. Lehmer [1] asked us to study 𝑁(𝑝) or at least to say something nontrivial about it. It is known that 𝑁(𝑝)≡2or0mod4 when 𝑝≡±1mod4. Some works related to the Lehmer problem can be found in references [2–5]. For example, Zhang [2, 4] proved the asymptotic formula 𝑁𝑝=12⋅𝑝+𝑂𝑝^1/2⋅ln²𝑝. (1) In this paper, we will study a new summation related to the Lehmer problem. For any fixed integer 𝑘≥2 and integer 𝑟 with (𝑟,𝑝)=1, we define the sums 𝑁(𝑘,𝑟;𝑝) as follows: 𝑁=1𝑘,𝑟;𝑝2^𝑝−1_𝑎1=1^𝑝−1_𝑎2=1^𝑝−1_{𝑎𝑘𝑎=1}⋯_{1𝑎2⋯𝑎𝑘≡𝑟mod𝑝}1−(−1)^{𝑎1+𝑎2+⋯+𝑎𝑘}. (2) 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 𝑁=12,1;𝑝2^𝑝−1_𝑎=1^𝑝−1_{𝑏=1𝑎𝑏≡1mod𝑝}1−(−1)^𝑎+𝑏=12^𝑝−1_𝑎=11−(−1)^𝑎+𝑎=𝑁𝑝. (3) 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 𝑘≥2 and integer 𝑟 with (𝑟,𝑝)=1, we have the asymptotic formula 𝑁=1𝑘,𝑟;𝑝2⋅𝑝−1^𝑘−1+𝑂𝑝^(𝑘−1)/2⋅ln^𝑘𝑝. (4)
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 𝑞≥3 be an integer. For any integer 𝑚, we define 𝐶=𝑘,𝑚;𝑞_{𝑞𝑎1𝑎=11𝑞,𝑞=1𝑎2𝑎=12,𝑞=1𝑞𝑎𝑘=1(𝑎𝑘,𝑞)=1}⋯𝑒𝑎₁+𝑎₂+⋯+𝑎_𝑘+𝑚𝑎₁𝑎₂⋯𝑎_𝑘𝑞, (5) where 𝑒(𝑦)=𝑒^{2𝜋𝑖𝑦},𝑎_𝑖⋅𝑎_𝑖≡1mod𝑝,𝑖=1,2,…,𝑘.
About some arithmetical properties of 𝐶(𝑘,𝑚;𝑞), one can find them in [6–8]. 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 𝑘≥2, we have the asymptotic formula ^𝑝−1_𝑟=1𝐶𝑘−1,𝑟2^𝑘4;𝑝⋅𝐸𝑘,𝑟;𝑝=−^𝑘−1⋅𝑖^𝑘𝜋^𝑘⋅𝑝^𝑘+𝑂𝑝^{𝑘−1+𝜖}, (6) 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 ^𝑝−1_𝑟=1𝐶2,8𝑟;𝑝⋅𝐸3,𝑟;𝑝=16⋅𝑖𝜋³⋅𝑝³+𝑂𝑝^2+𝜖. (7)

Corollary 4. Let 𝑝 be an odd prime. Then for any fixed positive number 𝜖>0, we have the asymptotic formula ^𝑝−1_𝑟=1𝐶3,16𝑟;𝑝⋅𝐸4,𝑟;𝑝=−64𝜋⁴⋅𝑝⁴+𝑂𝑝^3+𝜖. (8)

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 𝑘≥1 and any integer 𝑚, we have the estimate 𝐶=𝑘,𝑚;𝑝_{𝑝−1𝑎1=1𝑝−1𝑎2=1}⋯^𝑝−1_𝑎𝑘=1𝑒𝑎₁+𝑎₂+⋯+𝑎_𝑘+𝑚𝑎₁𝑎₂⋯𝑎_𝑘𝑝≪𝑝^𝑘/2. (9)

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 ^𝑝−1_𝑎=1(−1)^𝑎(𝑎)(2)⋅1⋅𝜒=21−2𝜒𝑝⋅^𝑝−1_𝑎=1𝑎⋅𝜒(𝑎). (10)

Proof. See [10] or Lemma  3 in [3].

Lemma 7. Let 𝑝 be an odd prime. Then for any integer (𝑟,𝑝)=1, we have 𝑁=1𝑘,𝑟;𝑝2⋅𝑝−1^𝑘−1−2^𝑘−1⋅𝑖^𝑘𝜋^𝑘𝑝−1×_{𝜒()−1=−1}𝜒(𝑟)1−2𝜒(2)^𝑘𝜏^𝑘𝜒𝐿^𝑘1,𝜒, (11) where ∑_{𝜒(−1)=−1} denotes the summation over all odd characters 𝜒mod𝑝, 𝜏(𝜒)=∑^𝑝−1_𝑎=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 𝑁=1𝑘,𝑟;𝑝2^𝑝−1_𝑎1=1^𝑝−1_𝑎2=1^𝑝−1_{𝑎𝑘𝑎=1}⋯_{1𝑎2⋯𝑎𝑘≡𝑟mod𝑝}1−(−1)^{𝑎1+𝑎2+⋯+𝑎𝑘}=12⋅𝑝−1^𝑘−1−12^𝑝−1_𝑎1=1^𝑝−1_𝑎2=1^𝑝−1_{𝑎𝑘𝑎=1}⋯_{1𝑎2⋯𝑎𝑘≡𝑟mod𝑝}(−1)^{𝑎1+𝑎2+⋯+𝑎𝑘}=12⋅𝑝−1^𝑘−1−12𝑝−1×_𝜒mod𝑝𝜒(𝑟)^𝑝−1_𝑎=1(−1)^𝑎⋅𝜒(𝑎)^𝑘≡12⋅𝑝−1^𝑘−1−𝐴𝑘,𝑝. (12) For any odd character 𝜒mod𝑝, from Theorems 12.11 and 12.20 of [9] we have 1𝑝⋅^𝑝−1_𝑎=1(𝑎)=𝑖𝑎⋅𝜒𝜋⋅𝜏𝜒⋅𝐿1,𝜒. (13) Note that, for any even character 𝜒mod𝑝, we have the identity ^𝑝−1_𝑎=1(−1)^𝑎⋅𝜒(𝑎)=0, (14) from (13) and Lemma 6 we have 𝐴=2𝑘,𝑝^𝑘−1⋅𝑖^𝑘𝜋^𝑘×∑𝑝−1_{𝜒()−1=−1}𝜒(𝑟)1−2𝜒(2)^𝑘𝜏^𝑘𝜒𝐿^𝑘1,𝜒. (15) Now Lemma 7 follows from (12) and (15).

Lemma 8. Let 𝑝 be an odd prime and 𝑘 a fixed integer with 𝑘≥2. Then for any nonprincipal character 𝜒mod𝑝 and any real numbers 𝑦≥𝑝³, we have the estimate _𝑛≤𝑦𝜒(𝑛)⋅𝑑_𝑘(𝑛)≪𝑦^{1−(1/2𝑘−1)}⋅√𝑝⋅ln𝑝. (16)

Proof. We use mathematical induction to prove this lemma. If 𝑘=2, then from the Pòlya-Vinogradov inequality we have _𝑛≤𝑦𝜒(𝑛)𝑑(𝑛)=_{𝑚𝑛≤𝑦}𝜒()=𝑚𝑛2_{√𝑛≤𝑦}𝜒(𝑛)_{𝑚≤𝑦/𝑛}𝜒(𝑚)−_{√𝑛≤𝑦}𝜒(𝑛)²≪√𝑦√𝑝ln𝑝=𝑦^1−(1/2)⋅√𝑝⋅ln𝑝. (17) Assume that the lemma holds for 𝑘=𝑟. That is, _𝑛≤𝑦𝜒(𝑛)⋅𝑑_𝑟(𝑛)≪𝑦^{1−(1/2𝑟−1)}⋅√𝑝⋅ln𝑝. (18) Then for 𝑘=𝑟+1, note that 𝑑_𝑟+1(𝑛)=∑_𝑠∣𝑛𝑑_𝑟(𝑠); applying estimate (18) and the Pòlya-Vinogradov inequality we have _𝑛≤𝑦𝜒(𝑛)𝑑_𝑟+1(𝑛)=_{𝑚𝑛≤𝑦}𝜒(𝑚𝑛)𝑑_𝑟(𝑚)≤_{√𝑛≤𝑦}𝜒(𝑛)_{𝑚≤𝑦/𝑛}𝜒(𝑚)𝑑_𝑟(𝑚)+_{√𝑚≤𝑦}𝜒(𝑚)𝑑_𝑟(𝑚)_{𝑛≤𝑦/𝑛}𝜒(𝑛)+_{√𝑛≤𝑦}𝜒(𝑛)⋅_{√𝑚≤𝑦}𝜒(𝑚)𝑑_𝑟(𝑚)≪_{√𝑛≤𝑦}𝑦𝑛^{1−(1/2𝑟−1)}⋅√𝑝⋅ln𝑝+𝑦^{1−(1/2𝑟)}⋅√𝑝⋅ln𝑝≪𝑦^{1−(1/2𝑟)}⋅√𝑝⋅ln𝑝. (19) 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 𝑁≥𝑝^𝑘, applying Abel’s identity (see Theorem 4.2 of [9]) we have 𝐿^𝑘1,𝜒=_{1≤𝑛≤𝑁}𝜒(𝑛)𝑑_𝑘(𝑛)𝑛+^∞_𝑁1𝑦²_{𝑁<𝑛≤𝑦}𝜒(𝑛)𝑑_𝑘(𝑛)𝑑𝑦. (20) For any integer 0≤𝑖≤𝑘, from Lemma 5 and the definition of 𝐶(𝑘,𝑚;𝑝) we have_{𝜒()−1=−1}𝜒(𝑟)𝜒2^𝑖𝜏^𝑘𝜒_{1≤𝑛≤𝑁}𝜒(𝑛)𝑑_𝑘(𝑛)𝑛=𝑝−12_{1≤𝑛≤𝑁𝑛,𝑝=1}𝑑_𝑘(𝑛)𝑛×^𝑝−1_𝑎1=1⋯^𝑝−1_{𝑎𝑘−1=1}𝑒𝑎₁+⋯+𝑎_𝑘−1+𝑛𝑟2^𝑖⋅𝑎₁𝑎₂⋯𝑎_𝑘−1𝑝−𝑝−12_{1≤𝑛≤𝑁𝑛,𝑝=1}𝑑_𝑘(𝑛)𝑛×^𝑝−1_𝑎1=1⋯^𝑝−1_{𝑎𝑘−1=1}𝑒𝑎₁+⋯+𝑎_𝑘−1−𝑛𝑟2^𝑖⋅𝑎₁𝑎₂⋯𝑎_𝑘−1𝑝≪𝑝−12_{1≤𝑛≤𝑁(𝑛,𝑝)=1}𝑑_𝑘(𝑛)𝑛⋅𝑝^(𝑘−1)/2≪𝑝^(𝑘+1)/2⋅ln^𝑘𝑁. (21)Applying (20) and the binomial expression we have the estimate _{𝜒()−1=−1}𝜒(𝑟)1−2𝜒(2)^𝑘𝜏^𝑘𝜒_{1≤𝑛≤𝑁}𝜒(𝑛)𝑑_𝑘(𝑛)𝑛≪𝑝^(𝑘+1)/2⋅ln^𝑘𝑁. (22) Taking 𝑁=𝑝^2𝑘−1, note that |𝜏(𝜒)|=√𝑝 and the identity _{𝑁<𝑛≤𝑦}𝜒(𝑛)𝑑_𝑘(𝑛)=_𝑛≤𝑦𝜒(𝑛)𝑑_𝑘(𝑛)−_𝑛≤𝑁𝜒(𝑛)𝑑_𝑘(𝑛), (23) and applying Lemma 8 we have the estimate _{𝜒()−1=−1}𝜒(𝑟)1−2𝜒(2)^𝑘𝜏^𝑘𝜒×^∞_𝑁1𝑦²_{𝑁<𝑛≤𝑦}𝜒(𝑛)𝑑_𝑘(𝑛)𝑑𝑦≪𝑝^(𝑘+2)/2⋅√𝑝𝑁^1/2𝑘−1⋅ln𝑝=𝑝^(𝑘+1)/2⋅ln𝑝. (24) Combining (20), (22), (24), and Lemma 7 we may immediately deduce the asymptotic formula 𝑁=1𝑘,𝑟;𝑝2⋅𝑝−1^𝑘−1+𝑂𝑝^(𝑘−1)/2⋅ln^𝑘𝑝. (25) 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^𝑝−1_𝑟=1𝜒(𝑟)⋅𝐶𝑘−1,𝑟2^𝑘=;𝑝_{𝑝−1𝑟=1}𝜒(𝑟)×^𝑝−1_𝑎1=1^𝑝−1_𝑎2=1⋯^𝑝−1_{𝑎𝑘−1=1}𝑒𝑎₁+𝑎₂+⋯+𝑎_𝑘−1+𝑟2^𝑘𝑎₁𝑎₂⋯𝑎_𝑘−1𝑝=^𝑝−1_𝑎1=1⋯^𝑝−1_{𝑎𝑘−1=1}𝑒𝑎₁+𝑎₂+⋯+𝑎_𝑘−1𝑝×^𝑝−1_𝑟=1𝜒(𝑟)𝑒𝑟2^𝑘𝑎₁𝑎₂⋯𝑎_𝑘−1𝑝=𝜏𝜒𝜒2^𝑘×^𝑝−1_𝑎1=1⋯^𝑝−1_{𝑎𝑘−1=1}𝜒𝑎₁𝑎₂⋯𝑎_𝑘−1𝑒𝑎₁+𝑎₂+⋯+𝑎_𝑘−1𝑝=𝜒2^𝑘⋅𝜏^𝑘𝜒. (26)Note that 𝜏^𝑘(𝜒)⋅𝜏^𝑘(𝜒)=𝜒^𝑘(−1)𝜏(𝜒)^𝑘⋅𝜏^𝑘(𝜒)=𝜒^𝑘(−1)⋅𝑝^𝑘; from Lemma 7 and the definition of 𝐸(𝑘,𝑟;𝑝) we have ^𝑝−1_𝑟=1𝐶𝑘−1,𝑟2^𝑘2;𝑝⋅𝐸𝑘,𝑟;𝑝=−^𝑘−1⋅𝑖^𝑘𝜋^𝑘𝑝−1⋅𝑝^𝑘⋅_{𝜒()−1=−1}(−1)^𝑘1−2𝜒(2)^𝑘𝜒2^𝑘𝐿^𝑘1,𝜒=(−1)^𝑘+12^𝑘−1⋅𝑖^𝑘𝜋^𝑘𝑝−1⋅𝑝^𝑘⋅^𝑘_𝑗=0𝑗𝑘(−1)^𝑗⋅2^𝑗⋅_{𝜒()−1=−1}𝜒2^{𝑘−𝑗∞}_𝑛=1𝜒(𝑛)⋅𝑑_𝑘(𝑛)𝑛=(−1)^𝑘+12^𝑘−1⋅𝑖^𝑘𝜋^𝑘𝑝−1⋅𝑝^𝑘⋅𝑝−12(−1)^𝑘⋅2^𝑘+𝑂𝑝^{𝑘−1+𝜖}4=−^𝑘−1⋅𝑖^𝑘𝜋^𝑘⋅𝑝^𝑘+𝑂𝑝^{𝑘−1+𝜖}, (27) where 𝑖²=−1, ^𝑗_𝑘=𝑘!/(𝑗!⋅(𝑘−𝑗)!), 𝜖 denotes any fixed positive number and 𝑑_𝑘(𝑛) denotes the 𝑘th divisor function. That is, 𝑑_𝑘(𝑛)=(∑_𝑑1∣𝑛∑_𝑑2∣𝑛⋯∑_{𝑑𝑘∣𝑛})_{𝑑1𝑑2⋯𝑑𝑘=𝑛}1.

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.