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

and Applied Analysis 3


Introduction
Let  > 1 be an integer.For any integer  with (, ) = 1, from the Euler-Fermat theorem we know that  () ≡ 1 mod , where () denotes Euler function.Let  be the smallest positive integer such that   ≡ 1 mod .If  = (), then  is called a primitive root of .If  has a primitive root, then each reduced residue system mod  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  ≥ 1.
Many researchers focused on the properties of primitive roots and some related problems and have obtained many interesting results; see [1][2][3][4][5][6][7].For example, Moreno and Sotero [4] proved that Golomb's conjecture is true for all  < 2 60 .That is, there exist two primitive elements  and  in finite fields F  such that  +  = 1, if  < 2 60 .Cohen and Mullen [2] established a generalization of Golomb's conjecture by proving the existence of  0 > 0 such that, whenever  >  0 , there exist primitive ,  ∈ F  with  +  = , where , , and  are arbitrary nonzero members of F  .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  > 3 be an odd prime number.For any integer  with (, ) = 1, let (, ) denote the number of all solutions of the congruence equation  −  ≡  mod , where  and  are the primitive roots mod .We define (, ) = 0, if  ≡ 0 mod , 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.
We may immediately deduce the following corollary from this theorem.Corollary 2. Let  > 3 be a prime number.Then for any three integers , , and  with (, ) = ( 2 − 4, ) = 1, one has

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 (, ) = 1, one has the identity where ind  denotes the index of  relative to some fixed primitive root of ; () is the M ö bius function.

Lemma 5.
Let  be an odd prime and let  be an integer with (, ) = 1.Then one has the identity where (( ind )/ℎ) =  ,ℎ () is the Dirichlet character mod .
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 mod .
From formula (13) and the definition of (, ) we may immediately deduce Lemma 5.