Abstract and Applied Analysis

Volume 2013 (2013), Article ID 326597, 19 pages

http://dx.doi.org/10.1155/2013/326597

Research Article

On the Mean Values of Certain Character Sums

Zhefeng Xu and Huaning Liu

Department of Mathematics, Northwest University, Xi'an, Shaanxi 710069, China

Received 8 August 2013; Accepted 22 September 2013

Academic Editor: T. Diagana

Copyright © 2013 Zhefeng Xu and Huaning Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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)–(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

Note that and for even character . By Lemma 12 we have

Now assume that . Then . We have

Note that and for even character . By Lemma 12 we have

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

Theorem 14. Let be an even primitive character modulo odd integer . Then one has

Proof. By Lemma 13, (3), and (37), we have

Therefore

4. Mean Values of Dirichlet -Functions

In this section, we will study the mean values of Dirichlet -functions, which will be used to prove Theorems 3 and 4.

Lemma 15. Let and be integers with and . Then one has the identities

where denotes the summation over all primitive characters modulo , and is the number of primitive characters modulo .

Proof. This is Lemma 3 of [12].

Lemma 16. Let be an odd number, and let be an integer. Then one has

where , , and .

Proof. By the Euler product, we have

Similarly, we can deduce the other identities.

Lemma 17. Let be an odd number. For integers and , one has

Proof. We only prove the first formula since, similarly, we can get the others. Let be the divisor function. For , by Abel’s identity, we get

For , from Lemma 15, we have

Then from Lemma 16 we get

Now taking , we immediately get

Lemma 18. Let be an odd number. For integers and , one has

Proof. By Lemma 17 and the methods proving Lemma 18, we can get this lemma.

5. Proof of Theorems 3 and 4

First we prove Theorem 3. By Theorem 11 and Lemma 18, we have

This proves Theorem 3.

On the other hand, by Theorem 14 and Lemma 18, we have

This completes the proof of Theorem 4.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant no. 11001218, the Natural Science Foundation of Shaanxi Province of China under Grant no. 2013JM1017 and 2011JQ1010, and the Natural Science Foundation of the Education Department of Shaanxi Province of China under Grant no. 2013JK0558.

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