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Research Article | Open Access

Volume 2013 |Article ID 326597 | https://doi.org/10.1155/2013/326597

Zhefeng Xu, Huaning Liu, "On the Mean Values of Certain Character Sums", Abstract and Applied Analysis, vol. 2013, Article ID 326597, 19 pages, 2013. https://doi.org/10.1155/2013/326597

# On the Mean Values of Certain Character Sums

Accepted22 Sep 2013
Published27 Nov 2013

#### 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.  have proved that for and for . Fine  has showed that for , there exist infinitely many primes with and infinitely many with .

Williams  proved that for being the Legendre symbol modulo . For primitive character modulo , Toyoizumi  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  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 , 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  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  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)–(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  or ) 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