On the Mean Values of Certain Character Sums

Let 𝑞 ≥ 5 be an odd number. In this paper, we study the fourth power mean of certain character sums ∑ ∗𝜒 mod 𝑞,(𝜒−1)=−1 |∑ 1≤𝑎≤𝑞/4 𝑎𝜒(𝑎)| 4 and ∑ ∗𝜒 mod 𝑞,(𝜒−1)=1 |∑ 1≤𝑎≤𝑞/4 𝑎𝜒(𝑎)| 4 , where ∑ ∗ denotes the summation over primitive characters modulo 𝑞 , and give some asymptotic formulae.


Introduction
The sum appears frequently in number theory, where is a nonprincipal primitive character modulo , and has been studied by several experts. For example, for ≡ 3 mod 4 being a prime and being the Legendre symbol, Ayoub et al. [1] have proved that ( ) < 0 for = 1, 2 and for ≥ − 2. Fine [2] has showed that for > 2, there exist infinitely many primes ≡ 3 mod 4 with ( ) > 0 and infinitely many with ( ) < 0.
In [5], Peral used the Gauss sums and adequate Fourier expansion to greatly improve the result in Proposition 1.
It may be interesting to consider the mean value of certain character sums. For example, Burgess [7] proved that * where ∑ * denotes the summation over primitive characters modulo , and ( ) is the Dirichlet divisor function. Xu and Zhang studied the power mean * (11) in [8,9] and obtained some sharper results.
In this paper, we study the fourth power mean of certain character sums * ∑ mod and give a few asymptotic formulae.

Theorem 3. Let ≥ 5 be an odd number. Then one has
where ( ) is the number of primitive characters modulo , 4 is the nonprincipal character modulo 4, and is any fixed positive real number.

Theorem 4. Let ≥ 5 be an odd number. Then one has
Remark 7. It seems that the contributions of odd and even primitive characters to the fourth power moment of character sums over [1, /4] are very different.

Express the Character Sum in terms of Gauss Sums and -Functions (I)
Let be an odd primitive character modulo . In this section, we will express ∑ 1≤ ≤ /4 ( ) in terms of Gauss sums and Dirichlet -functions. We need the following lemmas.
Lemma 8. Suppose that ≥ 5 is an odd number, and is an odd character modulo .

Express the Character Sum in terms of Gauss Sums and -Functions (II)
Let be an even primitive character modulo . In this section, we express ∑ 1≤ ≤ /4 ( ) in terms of Gauss sums and Dirichlet -functions. (47) (2 − 1) (2 − 1) (2 − 1) (2 − 1) we have Abstract and Applied Analysis It is not hard to show that (51) Therefore
Proof. By the Euler product, we have (2 ) (2 + ) Similarly, we can deduce the other identities. (2 ) (1, ) Proof. We only prove the first formula since, similarly, we can get the others. Let ( ) = ∑ | 1 be the divisor function.
Proof. By Lemma 17 and the methods proving Lemma 18, we can get this lemma.

Proof of Theorems 3 and 4
First we prove Theorem 3. By Theorem 11 and Lemma 18, we have *