#### Abstract

Let be an odd number. In this paper, we study the fourth power mean of certain character sums and , where denotes the summation over primitive characters modulo , and give some asymptotic formulae.

#### 1. 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 being a prime and being the Legendre symbol, Ayoub et al. [1] have proved that for and for . Fine [2] has showed that for , there exist infinitely many primes with and infinitely many with .

Williams [3] proved that for being the Legendre symbol modulo . For primitive character modulo , Toyoizumi [4] used the generalized Bernoulli numbers to express in terms of Gauss sums and Dirichlet -functions as follows: where is the Gauss sum, , is the Dirichlet -function corresponding to , and denotes the binomial coefficient.

Toyoizumi [4] also gave explicit bounds for .

Proposition 1. *(a) Assume that and . Then for any primitive character , one has
**
where
**
and is the Riemann zeta function.**(b) Assume that and . Then for any primitive character , one has
**
where
*

In [5], Peral used the Gauss sums and adequate Fourier expansion to greatly improve the result in Proposition 1.

Proposition 2. * (a) Assume that is a primitive nonprincipal character modulo , and then
**(b) Assume that is a primitive character modulo ; then,
*

Furthermore, Liu and Zhang [6] gave an upper bound for when is a nonprincipal character modulo .

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 in [8, 9] and obtained some sharper results.

In this paper, we study the fourth power mean of certain character sums and give a few asymptotic formulae.

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

Theorem 4. * Let be an odd number. Then one has
*

From Theorems 3 and 4, we immediately get the following corollaries.

Corollary 5. * Let be a prime. Then one has
**
where denotes the product over all primes. *

Corollary 6. * Let be a prime. Then
*

*Remark 7. * It seems that the contributions of odd and even primitive characters to the fourth power moment of character sums over are very different.

#### 2. 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 in terms of Gauss sums and Dirichlet -functions. We need the following lemmas.

Lemma 8. * Suppose that is an odd number, and is an odd character modulo .*(i)* For , one has
*(ii)* For , one has
*

*Proof. *It is easy to show that
This proves (i). Similarly, we can deduce (ii).

Lemma 9. * Suppose that is an odd number, and is an odd character modulo . Let be the nonprincipal character modulo 4. For , one has
*

*Proof. *Note that and , and we get

First we have
where is the inverse of modulo with and . Since , we get . Then from Lemma 8, we have
Therefore

On the other hand, we get

Since and , we have

Note that
so we get

Now combine (21)â€“(28); we have

Lemma 10. * Suppose that is an odd number, and is an odd character modulo . Let be the nonprincipal character modulo . For , one has
*

*Proof. *For , we get and . Using the methods of proving Lemma 9, we have

It is not hard to show that
Then by (32), we have

On the other hand, by Lemma 8, we get
Then from (33), we have

Combining (31), (35), and (37), we have

Now we can express in terms of Gauss sums and Dirichlet -functions.

Theorem 11. * Let be an odd primitive character modulo odd integer , and let be the nonprincipal character modulo . Then one has
*

*Proof. *By Lemmas 9 and 10, we get
From the Fourier expansion for primitive character sums (see [10] or [11])
we easily have

Note that is a primitive character modulo satisfying , and
then from (3) we have
Therefore
Then we have

#### 3. Express the Character Sum in terms of Gauss Sums and -Functions (II)

Let be an even primitive character modulo . In this section, we express in terms of Gauss sums and Dirichlet -functions.

Lemma 12. * Let be an odd number, and let be a nonprincipal character modulo . Then
*

*Proof. *We have

Since
we have

It is not hard to show that
Therefore
Note that
we have

Lemma 13. *Let be an odd number, and let be an nonprincipal even character modulo . If , then
**
While if , we have
*

*Proof. *First suppose that . Then . We have