On a Kind of Dirichlet Character Sums

Abstract

Let 𝑝≥3 be a prime and let 𝜒 denote the Dirichlet character modulo 𝑝. For any prime 𝑞 with 𝑞<𝑝, define the set 𝐸(𝑞,𝑝)={𝑎∣1≤𝑎,𝑎≤𝑝,𝑎𝑎≡1(mod𝑝)and𝑎≡𝑎(mod𝑞)}. In this paper, we study a kind of mean value of Dirichlet character sums ∑_{𝑎≤𝑝𝑎∈𝐸(𝑞,𝑝)}𝜒(𝑎), and use the properties of the Dirichlet 𝐿-functions and generalized Kloosterman sums to obtain an interesting estimate.

1. Introduction

Let 𝑘≥3 be an integer and let 𝜒 denote the Dirichlet character modulo 𝑘, for any real number 𝑥≥1, many scholars have studied the following sums: _𝑛≤𝑥𝜒(𝑛), (1) where 𝑛 are positive integers.

Perhaps one of the most famous results is Pólya’s inequality [1]. That is, when 𝜒 is the primitive character modulo 𝑘, we have _𝑛≤𝑥𝜒(𝑛)<𝑘^1/2log𝑘. (2)

In fact, the result can be extended to the nonprincipal character 𝜒 modulo 𝑘 [2]. Further details about the estimates of character sums can be found in the literature, for example, [3, 4].

For any fixed integer 𝐻>0 and any positive integer 𝑘≥3, define the following set: 𝐿(𝐻,𝑘)={𝑎∣1≤𝑎,()𝑎≤𝑘−1,𝑎,𝑘=1,𝑎𝑎≡1(mod𝑘),|𝑎−𝑎|≤𝐻}, (3) let 𝜒 denote the Dirichlet character modulo 𝑘, define the sums as follows: _{𝑛≤𝑘𝑛∈𝐿(𝐻,𝑘)}𝜒(𝑛). (4)

Xi and Yi [5] studied the problem for 𝜒 the nonprincipal Dirichlet character modulo 𝑘, and got _{𝑛≤𝑘𝑛∈𝐿(𝐻,𝑘)}𝜒(𝑛)≪𝑘^1/2𝑑(𝑘)log𝐻, (5) where 0<𝐻≤𝑞 was a constant and 𝑑(𝑘) was the divisor function. Before this, Wenpeng [6] got an asymptotic formula for the case that 𝜒 was the principal Dirichlet character modulo 𝑘.

On the other hand, for each integer 𝑎 with 1≤𝑎≤𝑘 and (𝑎,𝑘)=1, we know that there exists one and only one 𝑏 with 1≤𝑏≤𝑘 such that 𝑎𝑏≡1(mod𝑘). Let 𝑟₂(𝑘) be the number of solutions of the congruent equation 𝑎𝑏≡1(mod𝑘) for 1≤𝑎, 𝑏≤𝑘 in which 𝑎 and 𝑏 are of opposite parity, this can be expressed as follows: 𝑟₂(𝑘)=^𝑘_{𝑎𝑎=1()𝑎≡1mod𝑘2†(𝑎+𝑎)}1. (6)

Richard [7] asks us to find 𝑟₂(𝑘) or at least to say something nontrivial about it. About this problem, a lot of scholars have studied it [8–12]. Now we let 𝑚 be another integer with 𝑚<𝑘 and let 𝑟_𝑚(𝑘) denote the number of all pairs of integers 𝑎, 𝑏 satisfying 𝑎𝑏≡1(mod𝑘), 1≤𝑎, 𝑏≤𝑘, and 𝑚†(𝑎+𝑏). Lu and Yi [13] have obtained the asymptotic formula of generalized D. H. Lehmer problem as follows: 𝑟_𝑚(𝑘)=^𝑘_{𝑎𝑎=1()𝑎≡1mod𝑘𝑚†(𝑎+𝑎)}11=1−𝑚𝜙(𝑘)+𝑂𝑘^1/2log²𝑘, (7) where the 𝑂 constant only depends on 𝑚.

In this paper, let 𝑝 be an odd prime and let 𝑞 be a fixed prime with 𝑞<𝑝, define the set 𝐸(𝑞,𝑝) for 𝑎(1≤𝑎≤𝑝) such that 𝑎𝑎≡1(mod𝑝) and 𝑎≡𝑎(mod𝑞), that is, 𝐸𝑞,𝑝=𝑎∣1≤𝑎,𝑎𝑎≤𝑝−1,𝑎≡1mod𝑝,𝑎≡𝑎mod𝑞. (8)

As another case of (7), we will consider the mean value of Dirichlet character sums as follows: _{𝑎≤𝑝−1𝑎∈𝐸(𝑞,𝑝)}𝜒(𝑎), (9) and get an interesting estimate. That is, we will prove the following theorem.

Theorem 1. Let 𝑝 be an odd prime and let 𝑞 be a fixed prime with 𝑞<𝑝, and let 𝜒 denote the Dirichlet character modulo 𝑝. Let 𝐸(𝑞,𝑝) denote the following set: 𝐸=𝑎𝑞,𝑝𝑎∣1≤𝑎≤𝑝−1,𝑎≡1mod𝑝,𝑎≡𝑎mod𝑞, (10) then, for any nonprincipal Dirichlet character 𝜒mod𝑝, we have the following estimate: _{𝑎≤𝑝−1𝑎∈𝐸(𝑞,𝑝)}𝜒(𝑎)=𝑂𝑝^1/2+𝜖, (11) where the 𝑂 constant only depends on 𝑞.
From this Theorem we can get ^𝑝−1_{𝑎𝑎=1(𝑎≡1mod𝑝𝑞†𝑎−𝑎)}𝜒(𝑎)=^𝑝−1_{𝑎𝑎=1𝑎≡1(mod𝑝)}𝜒(𝑎)−_{𝑎≤𝑝−1𝑎∈𝐸𝑞,𝑝}𝜒(𝑎)=𝑂𝑝^1/2+𝜖. (12)
For any integer 𝑘 and fixed integer 𝑚 such that (𝑚,𝑘)=1, whether or not there exists an estimate for _{𝑎≤𝑘𝑎∈𝐸(𝑚,𝑘)}𝜒(𝑎) (13) is still an open problem.

2. Some Lemmas

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

Lemma 2. Let 𝑄 be an integer, and let 𝜒 be a primitive character modulo 𝑄. Then, for any real number 𝑢 and 𝑣 with 𝑢<𝑣, we have _{𝑢𝑄<𝑛≤𝑣𝑄}𝜒(𝑛)=𝜏𝜒_{|ℎ|0<≤𝐻}𝜒(ℎ)𝑒(−ℎ𝑢)−𝑒(−ℎ𝑣)2𝜋𝑖ℎ+𝑂1+𝑄log𝑄𝐻, (14) where 𝑒(𝑥)=𝑒^{2𝜋𝑖𝑥} and 𝜏(𝜒)=∑^𝑄_𝑎=1𝜒(𝑎)𝑒(𝑎/𝑄) are Gauss sums.
Especially, let 𝑢=0, we have a slight modification _{0<𝑛≤𝑣𝑄}𝜒(𝑛)=𝜏𝜒𝜋^∞_𝑛∑₌₁𝜒(𝑛)sin(2𝜋𝑛𝑣)𝑛(1),()𝜏𝜒+𝑂𝑖𝑓𝜒−1=1,𝜋𝑖^∞_𝑛∑₌₁𝜒(𝑛)(1−cos(2𝜋𝑛𝑣))𝑛(1),()+𝑂𝑖𝑓𝜒−1=−1. (15)

Proof. (See [1]).
 

Lemma 3. Let 𝑞 be a prime, let 𝑄 be an integer with 𝑄>𝑞, and let 𝜒 be a primitive character modulo 𝑄, then, we have _{𝑛≤𝑄/𝑞}𝜒(𝑛)=𝜏𝜒𝑞−1𝜋^𝑞−1_𝑙∑₌₁∑_𝜒2mod𝑞𝜒₂(𝑙)sin2𝜋𝑙𝑞×𝐿1,𝜒𝜒₂(1)()𝜏𝜒+𝑂,𝜒−1=1,𝑞−1𝜋𝑖^𝑞−1_𝑙∑₌₁∑_𝜒2mod𝑞𝜒₂(𝑙)1−cos2𝜋𝑙𝑞×𝐿1,𝜒𝜒₂+𝑂(1),𝜒(−1)=−1, (16) where 𝐿(1,𝜒) are the Dirichlet 𝐿-functions corresponding to 𝜒.

Proof. From Lemma 2, we take 𝑢=0, and 𝑣=1/𝑞 and get _{𝑛≤𝑄/𝑞}𝜒(𝑛)=𝜏𝜒_0<ℎ≤𝐻𝜒(ℎ)1−𝑒−ℎ/𝑞𝜒2𝜋𝑖ℎ+𝜏_{−𝐻<ℎ≤0}𝜒(ℎ)1−𝑒−ℎ/𝑞2𝜋𝑖ℎ+𝑂1+𝑄log𝑄𝐻. (17)
When 𝜒(−1)=1, we have _{𝑛≤𝑄/𝑞}𝜒(𝑛)=𝜏𝜒_0<ℎ≤𝐻𝜒(ℎ)sin2𝜋ℎ/𝑞𝜋ℎ+𝑂1+𝑄log𝑄𝐻=𝜏𝜒𝜋^𝑞−1_𝑙=1sin2𝜋𝑙𝑞_{0<ℎ≤𝐻ℎ≡𝑙mod𝑞}𝜒(ℎ)ℎ+𝑂1+𝑄log𝑄𝐻=𝜏𝜒𝑞−1𝜋^𝑞−1_𝑙=1sin2𝜋𝑙𝑞_0<ℎ≤𝐻𝜒(ℎ)ℎ×_𝜒2mod𝑞𝜒₂(ℎ)𝜒₂(𝑙)+𝑂1+𝑄log𝑄𝐻=𝜏𝜒𝑞−1𝜋^𝑞−1_{𝑙=1𝜒2mod𝑞}𝜒₂(𝑙)sin2𝜋𝑙𝑞×_0<ℎ≤𝐻𝜒𝜒₂(ℎ)ℎ+𝑂1+𝑄log𝑄𝐻. (18)
Let 𝐻→∞, then, we have _{𝑛≤𝑄/𝑞}𝜒(𝑛)=𝜏𝜒𝑞−1𝜋^𝑞−1_{𝑙=1𝜒2mod𝑞}𝜒₂(𝑙)sin2𝜋𝑙𝑞𝐿1,𝜒𝜒₂+𝑂(1). (19)
The case of 𝜒(−1)=−1 can be treated in the same way. This proves Lemma 3.

Lemma 4. Let 𝑝,𝑞 be odd primes and let 𝜒₁,𝜒₂ be the Dirichlet characters modulo 𝑝 and 𝑞, respectively, such that (𝑝,𝑞)=1, denote 𝜒=𝜒₁𝜒₂, 𝑘=𝑝𝑞, and 𝜒 is the Dirichlet character modulo 𝑝𝑞, the famous Gauss sums are defined as follows: 𝐺𝑛,𝜒=^𝑘_𝑎=1𝜒(𝑎)𝑒𝑛𝑎𝑘, (20) where 𝑒(𝑦)=𝑒^{2𝜋𝑖𝑦}. Hence, we have 𝐺𝑛,𝜒=𝜒₁𝑞𝜒₂𝑝𝐺𝑛,𝜒₁𝐺𝑛,𝜒₂. (21)
When 𝑛=1, we denote 𝜏(𝜒)=𝐺(1,𝜒); therefore, we have 𝜏𝜒=𝜒₁𝑞𝜒₂𝑝𝜏𝜒₁𝜏𝜒₂. (22)

Proof. (See [14]).
 

Lemma 5. Let 𝑚, and 𝑛 be integers and let 𝑞≥3 be prime, let 𝜒 denote the Dirichlet character modulo 𝑞, the generalized Kloosterman sums are defined by 𝑆_𝜒𝑚,𝑛;𝑞=_𝑎mod𝑞𝜒(𝑎)𝑒𝑚𝑎+𝑛𝑎𝑞, (23) where 𝑎𝑎≡1(mod𝑞) and 𝑒(𝑦)=𝑒^{2𝜋𝑖𝑦}.
Then, we have the following estimate: 𝑆_𝜒𝑚,𝑛;𝑞≪𝑞^1/2+𝜖𝑚,𝑛,𝑞^1/2, (24) where (𝑚,𝑛,𝑞) denotes the gcd of 𝑚,𝑛, and 𝑞.

Proof. (See [15]).
 

Lemma 6. Let 𝑝≥3 be an odd prime, let 𝜒, 𝜒₁ be a Dirichlet character modulo 𝑝, and 𝜒𝜒₁≠𝜒^𝑝₀. For any odd prime 𝑞 with 𝑞<𝑝, let 𝜒₂,𝜒₃,𝜒₄ be any Dirichlet characters with 𝜒₂mod𝑞, 𝜒₃mod𝑞 and 𝜒₄mod𝑞, respectively, then, no matter 𝜒 is odd character or even character modulo 𝑝, we have _{𝜒1≠𝜒0𝑝𝜒1𝜒2()−1=1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₂𝜒₃𝐿1,𝜒₁𝜒₂𝜒₄≪𝑝^3/2+𝜖;_{𝜒1≠𝜒0𝑝𝜒1𝜒2(−1)=−1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₂𝜒₃𝐿1,𝜒₁𝜒₂𝜒₄≪𝑝^3/2+𝜖;_{𝜒1≠𝜒0𝑝𝜒(−1)=1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₃𝐿1,𝜒₁𝜒₄≪𝑝^3/2+𝜖;_{𝜒1≠𝜒0𝑝𝜒()−1=−1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₃𝐿1,𝜒₁𝜒₄≪𝑝^3/2+𝜖, (25)(26)(27)(28) where the ≪ constant only depends on 𝑞.

Proof. For any integer 𝑛 with (𝑛,𝑘)=1 (𝑘≥3 is any positive integer), we have _{𝜒mod𝑘𝜒(−1)=1}𝜒(𝑛)=12𝜙(𝑘),𝑛≡±1(mod𝑘),0,otherwise._{𝜒mod𝑘𝜒(−1)=−1}𝜒(𝑛)=12𝜙(𝑘)(),−1,𝑛≡1mod𝑘2𝜙(𝑘),𝑛≡−1(mod𝑘),0,otherwise. (29)
Now let 𝑦>𝑝 and let 𝐴(𝑦,𝜒)=∑_{𝑝<𝑛≤𝑦}𝜒(𝑛). Then, from the Pólya-Vinogradov inequality, we obtain 𝐴𝑦,𝜒≪√𝑝log𝑝. (30)
Hence, from Abel’s identity, for any Re(𝑠)≥1, we can easily get 𝐿𝑠,𝜒=_𝑛≤𝑝𝜒(𝑛)𝑛^𝑠+^∞_𝑝𝐴𝑦,𝜒𝑦^𝑠+1𝑑𝑦=_𝑛≤𝑝𝜒(𝑛)𝑛^𝑠+𝑂log𝑝𝑝^𝑠−1/2=_𝑛≤𝑝𝜒(𝑛)𝑛^𝑠+𝑂log𝑝𝑝^1/2. (31) We will take (25), for example, to prove this lemma. For (𝑞,𝑝)=1, from the definition of 𝜏(𝜒) and (31), since 𝜒₁ is not the principle character modulo 𝑝 and 𝜒𝜒₁ is not the principle character modulo 𝑝, we have _{𝜒1≠𝜒0𝑝𝜒1𝜒2()−1=1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₂𝜒₃𝐿1,𝜒₁𝜒₂𝜒₄=_{𝜒1𝜒mod𝑝1𝜒2(−1)=1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×_𝑘≤𝑝𝜒𝜒₁𝜒₂𝜒₃(𝑘)𝑘+𝑂log𝑝𝑞(𝑝𝑞)^1/2×_𝑛≤𝑝𝜒₁𝜒₂𝜒₄(𝑛)𝑛+𝑂log𝑝𝑞𝑝𝑞^1/2−𝜒𝑞𝜏𝜒𝜏𝜒^𝑝₀𝐿1,𝜒𝜒₂𝜒₃𝐿1,𝜒₂𝜒₄=_{𝜒1𝜒mod𝑝1𝜒2(−1)=1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×_𝑘≤𝑝𝜒𝜒₁𝜒₂𝜒₃(𝑘)𝑘_𝑛≤𝑝𝜒₁𝜒₂𝜒₄(𝑛)𝑛+𝑂𝑝^3/2log²𝑝=_{𝑘≤𝑝𝑛≤𝑝}𝜒𝜒₁𝜒₃(𝑘)𝜒₂𝜒₄(𝑛)𝜒𝑞𝑎×𝑘𝑛_{𝑝′𝑝𝑎=1′𝑏=1}𝜒(𝑎)𝑒𝑎+𝑏𝑝×_{𝜒1𝜒mod𝑝1𝜒2()−1=1}𝜒₁𝑞²𝑎𝑏𝜒₁(𝑘𝑛)+𝑂𝑝^3/2log²𝑝=𝜙𝑝2_{𝑘≤𝑞𝑛≤𝑝}𝜒𝜒₁𝜒₃(𝑘)𝜒₂𝜒₄(𝑛)𝜒𝑞𝑎×𝑘𝑛_{𝑝′𝑝𝑎=1′𝑞𝑏=12𝑎𝑏≡𝑘𝑛mod𝑝}𝜒(𝑎)𝑒𝑎+𝑏𝑝±^𝑝′𝑝_𝑎=1^′_{𝑞𝑏=12𝑎𝑏≡−𝑘𝑛mod𝑝}𝜒(𝑎)𝑒𝑎+𝑏𝑝+𝑂𝑝^3/2log²𝑝=𝜙𝑝2_{𝑘≤𝑝𝑛≤𝑝}𝜒𝜒₁𝜒₃(𝑘)𝜒₂𝜒₄(𝑛)𝜒𝑞𝑎×𝑆𝑘𝑛_𝜒1,𝑞²𝑘𝑛;𝑝±𝑆_𝜒1,−𝑞²𝑝𝑘𝑛;𝑝+𝑂^3/2log²𝑝, (32) where 𝑎𝑏𝑞²𝑘𝑛≡1(mod𝑝). From Lemma 5, we can easily obtain _{𝜒1≠𝜒0𝑝𝜒1𝜒2()−1=1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁𝐿1,𝜒𝜒₁𝜒₂𝜒₃×𝐿1,𝜒₁𝜒₂𝜒₄≪𝑝^3/2+𝜖1log²𝑝≪𝑝^3/2+𝜖, (33) where the ≪ constant only depends on 𝑞. Therefore, this completes the proof of Lemma 6.

Lemma 7. Let 𝑞 be a fixed odd prime and let 𝑝 be a prime with 𝑝>𝑞, let 𝜒 denote the nonprincipal Dirichlet character modulo 𝑝, and 𝜒₁ denote the Dirichlet character modulo 𝑝, then, we have _{𝜒1𝜒mod𝑝1≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞×_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏)≪𝑝^3/2+𝜖, (34) where the ≪ constant only depends on 𝑞.

Proof. For primes 𝑝 and 𝑞, 𝜒₁ and 𝜒𝜒₁ are nonprincipal and primitive characters modulo 𝑝, hence, from Lemmas 3 and 6, when 𝜒(−1)=1, we can get _{𝜒1𝜒mod𝑝1≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏)=_{𝜒1()𝜒−1=11≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞×𝜏𝜒𝜒₁𝑞−1𝜋^𝑞−1_{𝑢=1𝜒3mod𝑞}𝜒₃(𝑢)sin2𝜋𝑢𝑞×𝐿1,𝜒𝜒₁𝜒₃(1)×𝜏𝜒+𝑂₁𝑞−1𝜋^𝑞−1_{𝑣=1𝜒4mod𝑞}𝜒₄(𝑣)sin2𝜋𝑣𝑞×𝐿1,𝜒₁𝜒₄+𝑂(1)+_{𝜒1()𝜒−1=−11≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞×𝜏𝜒𝜒₁𝑞−1𝜋𝑖^𝑞−1_{𝑢=1𝜒3mod𝑞}𝜒₃(𝑢)1−cos2𝜋𝑢𝑞×𝐿1,𝜒𝜒₁𝜒₃(1)×𝜏𝜒+𝑂₁𝑞−1𝜋𝑖^𝑞−1_{𝑣=1𝜒4mod𝑞}𝜒₄(𝑣)1−cos2𝜋𝑣𝑞×𝐿1,𝜒₁𝜒₄(1)=1+𝑂𝑞−1²𝜋²×^𝑞−1_𝑢=1^𝑞−1_{𝑣=1𝜒3mod𝑞𝜒4mod𝑞}𝜒₃(𝑢)𝜒₄(𝑣)sin2𝜋𝑢𝑞sin2𝜋𝑣𝑞×_{𝜒1≠𝜒0𝑝𝜒1()−1=1}𝜒𝜒₁𝑞𝜒₁(𝑚)𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₃𝐿1,𝜒₁𝜒₄+−1𝑞−1²𝜋²×^𝑞−1_𝑢=1^𝑞−1_{𝑣=1𝜒3mod𝑞𝜒4mod𝑚}𝜒₃(𝑢)𝜒₄(𝑣)×1−cos2𝜋𝑢𝑞1−cos2𝜋𝑣𝑞×_{𝜒1≠𝜒0𝑝𝜒1()−1=−1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₃𝐿1,𝜒₁𝜒₄+𝑂𝑝^1/2+𝜖≪𝑝^3/2+𝜖, (35) where the ≪ constant is only concerned with 𝑞.
When 𝜒(−1)=−1, by the similar method, we can also obtain _{𝜒1𝜒mod𝑝1≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏)≪𝑝^3/2+𝜖, (36) where the ≪ constant is only concerned with 𝑞. Combining (35) and (36), we can obtain Lemma 7. This completes the proof of Lemma 7.

Lemma 8. Let 𝑞 be a fixed odd prime and let 𝑝 be a prime with 𝑝>𝑞, let 𝜒 denote the nonprincipal Dirichlet character modulo 𝑝, let 𝜒₁,𝜒₂ denote the Dirichlet character modulo 𝑝,𝑞 respectively, then, we have _{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁𝜒₂(𝑎)𝜒₁𝜒₂(𝑏)≪𝑝^3/2+𝜖, (37) where the ≪ constant is only concerned with 𝑞.

Proof. According to the properties of Dirichlet character, we can get _{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁𝜒₂(𝑎)𝜒₁𝜒₂(𝑏)=_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1}𝜒𝜒₁𝜒₂(𝑎)−_{𝑎≤𝑝−1𝑞∣𝑎}𝜒𝜒₁𝜒₂(𝑎)×_{𝑏≤𝑝−1}𝜒₁𝜒₂(𝑏)−_{𝑏≤𝑝−1𝑞∣𝑏}𝜒₁𝜒₂(𝑏)=_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1}𝜒𝜒₁𝜒₂(𝑎)−𝜒𝜒₁𝜒₂𝑞×_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁𝜒₂(𝑎×_{𝑏≤𝑝−1})𝜒₁𝜒₂(𝑏)−𝜒₁𝜒₂𝑞_{𝑏≤(𝑝−1)/𝑞}𝜒₁𝜒₂(𝑏)=_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1}𝜒𝜒₁𝜒₂(𝑎)_{𝑏≤𝑝−1}𝜒₁𝜒₂(𝑏). (38)
For primes 𝑝 and 𝑞, let 𝜒^′=𝜒𝜒₁ be a nonprincipal and primitive character modulo 𝑝, 𝜒₂ is also a primitive character modulo 𝑞, so 𝜒𝜒₁𝜒₂ is a primitive character modulo 𝑝𝑞; therefore, from Lemmas 3, 4, and 6 and from (38), it is clear that when 𝜒(−1)=1, we can obtain _{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁𝜒₂(𝑎)𝜒₁𝜒₂(𝑏)=_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝜒1𝜒2(−1)=1}×𝜏𝜒𝜒₁𝜒₂𝑞−1𝜋^𝑞−1_{𝑢=1𝜒3mod𝑞}𝜒₃(𝑢)sin2𝜋𝑢𝑞×𝐿1,𝜒𝜒₁𝜒₂𝜒₃(1)×𝜏𝜒+𝑂₁𝜒₂𝑞−1𝜋^𝑞−1_{𝑣=1𝜒4mod𝑞}𝜒₄(𝑣)sin2𝜋𝑣𝑞×𝐿1,𝜒₁𝜒₂𝜒₄(1)++𝑂_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝜒1𝜒2(−1)=−1}×𝜏𝜒𝜒₁𝜒₂𝑞−1𝜋𝑖^𝑞−1_{𝑢=1𝜒3mod𝑞}𝜒₃(𝑢)1−cos2𝜋𝑢𝑞×𝐿1,𝜒𝜒₁𝜒₂𝜒₃(1)×𝜏𝜒+𝑂₁𝜒₂𝑞−1𝜋𝑖^𝑞−1_{𝑣=1𝜒4mod𝑞}𝜒₄(𝑣)1−cos2𝜋𝑣𝑞×𝐿1,𝜒₁𝜒₂𝜒₄(1)=1+𝑂𝑞−1²𝜋^2𝑞−1_𝑢=1^𝑞−1_{𝑣=1𝜒2𝜒mod𝑞2≠𝜒0𝑞𝜒3mod𝑞𝜒4mod𝑞}×𝜒₂𝑝𝜏𝜒₂𝜒₂𝑝𝜏𝜒₂𝜒₃(𝑢)𝜒₄(𝑣)sin2𝜋𝑢𝑞×sin2𝜋𝑣𝑞_{𝜒1≠𝜒0𝑝𝜒1𝜒2()−1=1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₂𝜒₃𝐿1,𝜒₁𝜒₂𝜒₄+−1𝑞−1²𝜋^2𝑞−1_𝑢=1^𝑞−1_{𝑣=1𝜒2𝜒mod𝑞2≠𝜒0𝑞𝜒3mod𝑞𝜒4mod𝑞}×𝜒₂𝑝𝜏𝜒₂𝜒₂𝑝𝜏𝜒₂𝜒₃(𝑢)𝜒₄(𝑣)×1−cos2𝜋𝑢𝑞1−cos2𝜋𝑣𝑞×_{𝜒1≠𝜒0𝑝𝜒1𝜒2()−1=−1}𝜒𝜒₁𝑞𝜒₁𝑞𝜏𝜒𝜒₁𝜏𝜒₁×𝐿1,𝜒𝜒₁𝜒₂𝜒₃𝐿1,𝜒₁𝜒₂𝜒₄≪𝑝^3/2+𝜖, (39) where the ≪ constant is only concerned with 𝑞.
When 𝜒(−1)=−1, in the similar way, we can also obtain _{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁𝜒₂(𝑎)𝜒₁𝜒₂(𝑏)≪𝑝^3/2+𝜖, (40) where the ≪ constant is only concerned with 𝑞.
Therefore, from (39) and (40), we can easily get Lemma 8. This completes the proof of Lemma 8.

3. Proof of Theorem

In this section, we will complete the proof of the theorem. According to the orthogonality relation for character sums, we have _{𝑎≤𝑝𝑎∈𝐸𝑞,𝑝}𝜒(𝑎)=^𝑝−1_𝑏=1^𝑝−1_{𝑎=1𝑎≡𝑏(mod𝑞)𝑎𝑏≡1(mod𝑝)}𝜒(𝑎)=1𝑝−1_𝜒1mod𝑝^𝑝−1_𝑏=1^𝑝−1_{𝑎=1𝑎≡𝑏(mod𝑞)}𝜒𝜒₁(𝑎)𝜒₁(𝑏)=1𝑝−1^𝑝−1_𝑎=1^𝑝−1_{𝑏=1𝑎≡𝑏(mod𝑞)}𝜒(𝑎)+1𝑝−1_{𝜒1𝜒mod𝑝1≠𝜒0𝑝}^𝑝−1_𝑎=1^𝑝−1_{𝑏=1𝑎≡𝑏(mod𝑞)}𝜒𝜒₁(𝑎)𝜒₁(𝑏)=1𝑝−1^𝑝−1_𝑎=1𝜒(𝑎)^𝑝−1_{𝑏=1𝑏≡𝑎(mod𝑞)}1+1𝑝−1_{𝜒1𝜒mod𝑝1≠𝜒0𝑝}^𝑝−1_{𝑎=1𝑞∣𝑎}^𝑝−1_{𝑏=1𝑞∣𝑏}𝜒𝜒₁(𝑎)𝜒₁(𝑏)+1𝑝−1_{𝜒1𝜒mod𝑝1≠𝜒0𝑝}^𝑝−1_{𝑎=1𝑞†𝑎}^𝑝−1_{𝑏=1𝑞†𝑏𝑎≡𝑏(mod𝑞)}𝜒𝜒₁(𝑎)𝜒₁(𝑏)=1𝑝−1^𝑝−1_𝑎=1𝜒(𝑎)𝑝𝑞(1)+1+𝑂𝑝−1_{𝜒1𝜒mod𝑝1≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞×_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏)+1𝑝−1𝜙𝑞×_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2mod𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁(𝑎)𝜒₁(𝑏)×𝜒₂(𝑎)𝜒₂(𝑏)=𝑝𝑝−1𝑞^𝑝−1_𝑎=1𝜒(𝑎)(1)+1+𝑂𝑝−1_{𝜒1𝜒mod𝑝1≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞×_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏)+1𝑝−1𝜙𝑞×_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝑎≤𝑝−1𝑚†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁(𝑎)𝜒₁(𝑏)+1𝑝−1𝜙𝑞×_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁(𝑎)𝜒₁(𝑏)×𝜒₂(𝑎)𝜒₂(𝑏)(1)+1=𝑂𝑝−1_{𝜒1𝜒mod𝑝1≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞×_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏)+1𝑝−1𝜙𝑞×_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁(𝑎)𝜒₁(𝑏)+1𝑝−1𝜙𝑞×_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁𝜒₂(𝑎)𝜒₁𝜒₂(𝑏). (41)

Note that for any nonprincipal Dirichlet character 𝜒 modulo 𝑘 (𝑘≥3 is an positive integer), we have ∑^𝑘_𝑛=1𝜒(𝑛)=0, hence, we obtain _{𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁(𝑎)𝜒₁(𝑏)=_{𝑎≤𝑝−1}𝜒𝜒₁(𝑎)−_{𝑎≤𝑝−1𝑞∣𝑎}𝜒𝜒₁(𝑎×_{𝑏≤𝑝−1})𝜒₁(𝑏)−_{𝑏≤𝑝−1𝑞∣𝑏}𝜒₁(𝑏)=_{𝑎≤𝑝−1𝑞∣𝑎}𝜒𝜒₁(𝑎)_{𝑏≤𝑝−1𝑞∣𝑏}𝜒₁(𝑏)=𝜒𝜒₁𝑞𝜒₁𝑞_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏). (42) From (41) and (42) and Lemmas 7 and 8, we get _{𝑎≤𝑝𝑎∈𝐸(𝑞,𝑝)}𝜒(𝑎)=11𝑝−11+𝜙𝑞×_{𝜒1𝜒mod𝑝1≠𝜒0𝑝}𝜒𝜒₁𝑞𝜒₁𝑞_{𝑎≤(𝑝−1)/𝑞}𝜒𝜒₁(𝑎)_{𝑏≤(𝑝−1)/𝑞}𝜒₁(𝑏)+1𝑝−1𝜙𝑞×_{𝜒1𝜒mod𝑝1≠𝜒0𝑝𝜒2𝜒mod𝑞2≠𝜒0𝑞𝑎≤𝑝−1𝑞†𝑎𝑏≤𝑝−1𝑞†𝑏}𝜒𝜒₁𝜒₂(𝑎)𝜒₁𝜒₂(𝑏)+𝑂(1)≪𝑝^1/2+𝜖, (43) where the ≪ constant only depends on 𝑞. This completes the proof of Theorem.

Acknowledgments

The authors greatly appreciate the referees for their kind comments and suggestions. This work is supported by the Basic Research Fund of Northwestern Polytechnical University of P. R. China (JC201123).