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

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 a 1 a 2 ⋯ a k ≡ r mod p. Let N(k, r; p) denote the number of all (a 1, a 2, ⋯ a k) such that a 1 a 2 ⋯ a k ≡ r mod p and 2†(a 1 + a 2 + ⋯ + 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.


Introduction
Let be an odd number. For each integer with 1 ≤ ≤ −1, it is clear that there exists one and only one with 0 ≤ ≤ − 1 such that ⋅ ≡ 1 mod . Let ( ) denote the number of all 1 ≤ ≤ − 1 in which and are of opposite parity. Professor D. H. Lehmer [1] asked us to study ( ) or at least to say something nontrivial about it. It is known that ( ) ≡ 2 or 0 mod 4 when ≡ ±1 mod 4. Some works related to the Lehmer problem can be found in references [2][3][4][5]. For example, Zhang [2,4] proved the asymptotic formula In this paper, we will study a new summation related to the Lehmer problem. For any fixed integer ≥ 2 and integer with ( , ) = 1, we define the sums ( , ; ) as follows: In fact, ( , ; ) is a generalization of the Lehmer problem. For example, if = 2 and = 1, then from the definition of (2, 1; ) we have So (2, 1; ) becomes ( ), the Lehmer problem. Now we are concerned about the arithmetical properties of ( , ; ). 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 ( , ; ) and give two interesting asymptotic formulae for it. That is, we will prove the following.

2
The Scientific World Journal Theorem 1. Let be an odd prime. Then for any fixed integer ≥ 2 and integer with ( , ) = 1, we have the asymptotic formula In order to facilitate the description of Theorem 2, we need to give the definition of high-dimension Kloosterman sums ( , ; ). Let ≥ 3 be an integer. For any integer , we define where ( ) = 2 , ⋅ ≡ 1 mod , = 1, 2, . . . , . About some arithmetical properties of ( , ; ), one can find them in [6][7][8]. Let ( , ; ) = ( , ; ) − (1/2)( − 1) −1 denote the error term in the asymptotic formula of ( , ; ). As another main content of this paper, we will study the asymptotic properties of the hybrid mean value of ( − 1, 2 ; ) and ( , ; ) and also give a sharp asymptotic formula for it. That is, we will prove the following.

Theorem 2. Let be an odd prime. Then for any fixed integer
≥ 2, we have the asymptotic formula where 2 = −1, denotes any fixed positive number. The constants 2 in Theorem 2 cannot be omitted. Otherwise, the main term in Theorem 2 is zero. If = 3 and 4, then from Theorem 2 we can also deduce the following two corollaries.

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 highdimension 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 be an odd prime. Then for fixed integer ≥ 1 and any integer , we have the estimate Proof. See [6,7].
For any odd character mod , from Theorems 12.11 and 12.20 of [9] we have Note that, for any even character mod , we have the identity from (13) and Lemma 6 we have (15) Now Lemma 7 follows from (12) and (15).

Lemma 8.
Let be an odd prime and a fixed integer with ≥ 2. Then for any nonprincipal character mod and any real numbers ≥ 3 , we have the estimate Proof. We use mathematical induction to prove this lemma. If = 2, then from the Pòlya-Vinogradov inequality we have Assume that the lemma holds for = . That is, Then for = + 1, note that +1 ( ) = ∑ | ( ); applying estimate (18) and the Pòlya-Vinogradov inequality we have Now our lemma follows from the induction.

Proofs of the Theorems
In this section, we will prove our conclusions. First we prove Theorem 1. For any real number ≥ , applying Abel's identity (see Theorem 4.2 of [9]) we have For any integer 0 ≤ ≤ , from Lemma 5 and the definition of ( , ; ) we have The Scientific World Journal Applying (20) and the binomial expression we have the estimate Taking = 2 −1 , note that | ( )| = √ and the identity and applying Lemma 8 we have the estimate Combining (20), (22), (24), and Lemma 7 we may immediately deduce the asymptotic formula The proof of Theorem 1 is right. Now we prove Theorem 2. For any nonprincipal character mod , from the definition and properties of Gauss sums we have ( 1 2 ⋅ ⋅ ⋅ −1 ) ( 1 + 2 + ⋅ ⋅ ⋅ + −1 ) = (2 ) ⋅ ( ) .