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 getNow 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.