A Mean Value Related to Primitive Roots and Golomb’s Conjectures

Abstract

The main purpose of this paper is using the properties of Gauss sums and the estimate for character sums to study a mean value problem related to the primitive roots and the different forms of Golomb’s conjectures and propose an interesting asymptotic formula for it.

1. Introduction

Let be an integer. For any integer with , from the Euler-Fermat theorem we know that , where denotes Euler function. Let be the smallest positive integer such that . If , then is called a primitive root of . If has a primitive root, then each reduced residue system can be expressed as a geometric progression. This gives a powerful tool that can be used in problems involving reduced residue systems. Unfortunately, not all moduli have primitive roots. In fact primitive roots exist only for the following moduli: where is an odd prime and .

Many researchers focused on the properties of primitive roots and some related problems and have obtained many interesting results; see [1–7]. For example, Moreno and Sotero [4] proved that Golomb’s conjecture is true for all . That is, there exist two primitive elements and in finite fields such that , if . Cohen and Mullen [2] established a generalization of Golomb’s conjecture by proving the existence of such that, whenever , there exist primitive with , where , , and are arbitrary nonzero members of . What is more, they also gave an asymptotic formula for the number of solutions. But we think the error term is too big and can be improved. In order to verify our viewpoint, we take the mean value properties of the error term into account. By using the properties of Gauss sums and the estimate for character sums, we obtained a stronger asymptotic formula.

Let be an odd prime number. For any integer with , let denote the number of all solutions of the congruence equation , where and are the primitive roots . We define , if , and

In this paper, we give an interesting asymptotic formula for the mean value of . This problem is interesting, because it cannot only reveal the profound properties of Golomb’s conjecture and provide the distribution law of the error term , but it is also a generalization of the related contents.

Theorem 1. Let be a prime. Then for any three integers ,  , and with , one has the asymptotic formula where is Euler function, , and denotes the number of all distinct prime divisors of .
We may immediately deduce the following corollary from this theorem.

Corollary 2. Let be a prime number. Then for any three integers , , and with , one has

2. Several Lemmas

In this section, we provide several lemmas that will be necessary for the proof of our theorem. Throughout this paper, we used many properties of Dirichlet characters and Gauss sums, which can be found in [8]. Firstly, we have the following lemma.

Lemma 3. Let be an odd prime. Then for any integer with , one has the identity where denotes the index of relative to some fixed primitive root of ; is the Möbius function.

Proof. See Proposition 2.2 of [9].

Lemma 4. Let be an odd prime; , , and are three integers with . Then for any nonprincipal character , one has the identity where denotes the Legendre symbol.

Proof. Since any nonprincipal character is a primitive character , so from the properties of Gauss sums we conclude that where is the classical Gauss sums.
On the other hand, for any integer with , we have For any integer with , from [10] (Section 7.8, Theorem 8.2) we also have Therefore,
Then from (7), (8), (9), and (10) we deduce the identity This proves Lemma 4.

Lemma 5. Let be an odd prime and let be an integer with . Then one has the identity where is the Dirichlet character .

Proof. From the trigonometric identity, the properties of classical Gauss sums, and Lemma 3 we have where we used the properties , if is not a principal character .
From formula (13) and the definition of we may immediately deduce Lemma 5.

3. Proof of Theorem 1

In this section, we shall complete the proof of our theorem. First from Lemma 5 and the definition of we have where and denote the corresponding formula, respectively, in the summation.

Now we will estimate and in (14), respectively. It is clear that if and , then must be a nonprincipal character . So for any integer with , from Lemma 4 we have the identity Therefore, we have To estimate in (14), we write , where includes all the characters such that , and is the principal character ; includes all the characters such that . Now note that if is the principal character , then and , . This time, from identity (9) we have So from (17) and Lemma 4 we have Applying Lemma 4 and the estimate for Gauss sums we also have the estimate Combining (14), (18), and (19) we may immediately deduce the asymptotic formula where . This completes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N. S. F. (11371291, 61202437) of China.