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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 326597, 19 pages
On the Mean Values of Certain Character Sums
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.
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.
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
and is the Riemann zeta function.
(b) Assume that and . Then for any primitive character , one has where
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
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 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