On a Kind of Dirichlet Character Sums

and Applied Analysis 3 When χ(−1) = 1, we have


Introduction
Let ≥ 3 be an integer and let denote the Dirichlet character modulo , for any real number ≥ 1, many scholars have studied the following sums: where are positive integers. Perhaps one of the most famous results is Pólya's inequality [1]. That is, when is the primitive character modulo , we have In fact, the result can be extended to the nonprincipal character modulo [2]. Further details about the estimates of character sums can be found in the literature, for example, [3,4].
For any fixed integer > 0 and any positive integer ≥ 3, define the following set: let denote the Dirichlet character modulo , define the sums as follows: Xi and Yi [5] studied the problem for the nonprincipal Dirichlet character modulo , and got where 0 < ≤ was a constant and ( ) was the divisor function. Before this, Wenpeng [6] got an asymptotic formula for the case that was the principal Dirichlet character modulo .
On the other hand, for each integer with 1 ≤ ≤ and ( , ) = 1, we know that there exists one and only one with 1 ≤ ≤ such that ≡ 1(mod ). Let 2 ( ) be the number of solutions of the congruent equation ≡ 1( mod ) for 1 ≤ , ≤ in which and are of opposite parity, this can be expressed as follows: Richard [7] asks us to find 2 ( ) or at least to say something nontrivial about it. About this problem, a lot of scholars have studied it [8][9][10][11][12]. Now we let be another integer with < and let ( ) denote the number of all pairs of integers , satisfying ≡ 1(mod ), 1 ≤ , ≤ , 2 Abstract and Applied Analysis and †( + ). Lu and Yi [13] have obtained the asymptotic formula of generalized D. H. Lehmer problem as follows: where the constant only depends on . In this paper, let be an odd prime and let be a fixed prime with < , define the set ( , ) for (1 ≤ ≤ ) such that ≡ 1(mod ) and ≡ (mod ), that is, As another case of (7), we will consider the mean value of Dirichlet character sums as follows: and get an interesting estimate. That is, we will prove the following theorem.

Theorem 1.
Let be an odd prime and let be a fixed prime with < , and let denote the Dirichlet character modulo . Let ( , ) denote the following set: then, for any nonprincipal Dirichlet character mod , we have the following estimate: where the constant only depends on . From this Theorem we can get For any integer and fixed integer such that ( , ) = 1, whether or not there exists an estimate for is still an open problem.

Some Lemmas
In this section, we will give several lemmas which are necessary in the proof of the theorem.

Lemma 3.
Let be a prime, let be an integer with > , and let be a primitive character modulo , then, we have where (1, ) are the Dirichlet -functions corresponding to .
Proof. From Lemma 2, we take = 0, and V = 1/ and get Abstract and Applied Analysis 3 When (−1) = 1, we have Let → ∞, then, we have The case of (−1) = −1 can be treated in the same way. This proves Lemma 3.

Lemma 5. Let , and be integers and let ≥ 3 be prime, let denote the Dirichlet character modulo , the generalized Kloosterman sums are defined by
where ≡ 1(mod ) and ( ) = 2 . Then, we have the following estimate: where ( , , ) denotes the gcd of , , and .

Lemma 6.
Let ≥ 3 be an odd prime, let , 1 be a Dirichlet character modulo , and 1 ̸ = 0 . For any odd prime with < , let 2 , 3 , 4 be any Dirichlet characters with 2 mod , 3 mod and 4 mod , respectively, then, no matter is odd character or even character modulo , we have where the ≪ constant only depends on .
Proof. For any integer with ( , ) = 1 ( ≥ 3 is any positive integer), we have Abstract and Applied Analysis Now let > and let ( , ) = ∑ < ≤ ( ). Then, from the Pólya-Vinogradov inequality, we obtain Hence, from Abel's identity, for any Re( ) ≥ 1, we can easily get We will take (25), for example, to prove this lemma. For ( , ) = 1, from the definition of ( ) and (31), since 1 is not the principle character modulo and 1 is not the principle character modulo , we have where 2 ≡ 1(mod ). From Lemma 5, we can easily obtain where the ≪ constant only depends on . Therefore, this completes the proof of Lemma 6.

Lemma 7. Let be a fixed odd prime and let be a prime with
> , let denote the nonprincipal Dirichlet character modulo , and 1 denote the Dirichlet character modulo , then, we have where the ≪ constant only depends on .

Abstract and Applied Analysis 5
Proof. For primes and , 1 and 1 are nonprincipal and primitive characters modulo , hence, from Lemmas 3 and 6, when (−1) = 1, we can get where the ≪ constant is only concerned with . When (−1) = −1, by the similar method, we can also obtain where the ≪ constant is only concerned with . Combining (35) and (36), we can obtain Lemma 7. This completes the proof of Lemma 7.

Lemma 8.
Let be a fixed odd prime and let be a prime with > , let denote the nonprincipal Dirichlet character modulo , let 1 , 2 denote the Dirichlet character modulo , respectively, then, we have where the ≪ constant is only concerned with .
Proof. According to the properties of Dirichlet character, we can get Abstract and Applied Analysis For primes and , let = 1 be a nonprincipal and primitive character modulo , 2 is also a primitive character modulo , so 1 2 is a primitive character modulo ; therefore, from Lemmas 3, 4, and 6 and from (38), it is clear that when (−1) = 1, we can obtain where the ≪ constant is only concerned with . When (−1) = −1, in the similar way, we can also obtain where the ≪ constant is only concerned with . Therefore, from (39) and (40), we can easily get Lemma 8. This completes the proof of Lemma 8.

Proof of Theorem
In this section, we will complete the proof of the theorem. According to the orthogonality relation for character sums, we have