Abstract
We prove the polynomial analogues of some Liouville identities from elementary number theory. Consequently several sums defined over the finite fields are evaluated by combining the results obtained and some of the results from sums of reciprocals of polynomials over .
1. Introduction and Background
Let be a prime number, for some positive integer and the finite field with elements. It is well known that and the set of all integers share many similarities. Analogue of many results on has been proven for . Both rings are principal ideal domains, both have the property that the residue class ring of any nonzero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. For the ring , the units are and every nonzero integer is a multiple by a unit of a positive integer. Similarly, the units of are the nonzero scalars , and every nonzero polynomial in is a multiple by a unit of a monic polynomial. Thus, one may think that many other results, which hold for , have their analogues in the ring . This is indeed the case. For example, analogues of the little theorem of Fermat and Euler, Wilsons theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet’s theorem on primes in arithmetic progression and many more well-known theorems from elementary number theory have been proven true for , see [1].
In the nineteenth century, Joseph Liouville introduced a powerful new method into elementary number theory that allowed him to get many interesting identities. His approach was also used to solve well-known problems such as sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums, and many others. Most of these problems are still today a subject of mathematical research.
In this present paper, we propose to study the polynomial analogue of some identities of Liouville in Williams [2, chapter 3]. More precisely, if and represent respectively the number of divisors and the sum of divisors of , andLiouville proved the following.
Theorem 1 (Liouville). Let , , , and . SetIf satisfies thenwhere represents the integer n satisfying .
He also proved many other results related to Theorem 1.
The main goal of this paper is to evaluate some sums of polynomials over involving the polynomial analogue of Theorem 1. We also evaluate the average of some functions defined on the set of all monic polynomials over .
Throughout this paper, we let . We define , , and to be the subset of all monic polynomials in , the subset of all monic polynomials in of degree , and the subset of all monic polynomials in of degree less than , respectively. We also let and to be respectively the subset of all monic irreducible polynomials in of degree and the subset of all monic irreducible polynomials in of degree less than .
For , we denote the degree of by , the number of polynomials coprime to with by , the number of divisors of by and we set
Finally, we set if and if . It is well known (see [1]) that every , can be written uniquely in the form
where is a nonzero element in , are irreducible monic polynomials, for , and are nonnegative integers.
The following 2 propositions will be needed later. Their proofs are omitted here and can be found in [1].
Proposition 2. Let . If the prime decomposition of is given by
, then
Proposition 3.
2. Some Results about Sum of Reciprocals of Polynomials over
In this section, we give a closed formula for some sums of reciprocals of polynomials over . These results will be used in the subsequent sections to help prove some polynomial identities in .
We start with the following definitions. Letting , we define and as
Further, we set , and, for an integer , we define
The next 2 theorems are due to Carlitz [3]. The first one provides an explicit formula for and if . Details of this theorem can be also found in [4].
Theorem 4 (Carlitz [3]). For each , we have
The next theorem is initially due to Carlitz [3], but an alternative proof was given by Hicks et al. [5] in a most recent paper published in 2012.
Theorem 5 (Carlitz [3]). For , , we have
More results about sum of reciprocals of polynomials can be also found in [6]. We state here some of them. Let be the subset of all monic reducible polynomials in of degree less than and we denote by the product of all monic irreducible polynomials of degree in , i.e., S. Nelson [6] in 2016 has proved the following theorem.
Theorem 6 (Nelson [6]). With , , and defined as above, we have
We conclude this section with our main theorem that states the following.
Theorem 7 (main theorem). Let , , and . Assume and , where . DefineIf , thenand
Theorem 7 will be used in Section 4 to establish some polynomials identities in These identities are obtained by evaluating certain sums taken over the subset of all monic polynomials and monic irreducible polynomials.
3. Polynomial Analogue of Some Liouville’s Identities
Let and . We define the function as the number of divisors of congruent to modulo , i.e.,
The following theorem is the polynomial analogue of [2, theorem 3.8]. It describes some of the properties of for polynomials. The proof follows the same steps as the one for integers proved by Liouville in [2].
Lemma 8. For , we have
Proof. The proof is essentially the same as the integer case. We haveFor , we have For the second equality, we have
Let . The division algorithm theorem states that that there exists a unique set of polynomials satisfying , with or . We define
Remark 9. Since for , is the maximum between and , for , we have
Lemma 10 (see [2]). Let and . Then
Proof. We have
The next theorem is the polynomial analogue of [2, theorem 3.9].
Theorem 11. Let and . If and , then
Proof. For each , there exist a unique set of polynomials such that . Therefore, we have
Our last polynomial analogue result in this section deals with the function defined for as
Lemma 12. For any ,
Proof. We have
4. Proof of the Main Theorem
In this section, we prove the main Theorem 7 stated in Section 2. Some other sums are also evaluated. We need the following lemma.
Lemma 13 (see [4]). Let and . Then
Proof. We haveTherefore, by Theorem 4 the proof is complete.
4.1. Proof of the Main Theorem
Proof. We have andTherefore, we haveRewriting the summation in terms of and yieldsHence Lemmas 10 and 13 implyTherefore, The proof of (18) is similar and is a direct consequence of Lemma 13.
Corollary 14. If , then
Proof. If , then, for each , we have which forces by Euler theorem for polynomial; see [1]. Therefore,and the corollary follows from the main theorem.
4.2. Evaluation of Some Sums in
Theorem 15. Let and for .
If , then
Proof. Using the main theorem, we obtainTherefore, if , Theorem 5 implieswhich complete the proof of (43).
Theorem 16. For , we have
Proof. By the main theorem, we haveTherefore, if , then Theorem 6 implieswhich completes the proof of (46). The proof of (47) is similar and follows directly from the main theorem and Theorem 6 of Nelson.
Theorem 17. For , we have
Proof. Using the main theorem, we obtainTherefore, if , then Theorem 6 implieswhich complete the proof of (50). The proof of (51) is similar and follows directly from the main theorem and Theorem 6 of Nelson.
Example 18. In this example, we let runs through the set of all linear polynomials.
Let and . If and , then
5. Conclusion
In this project, we prove the polynomial analogue of some Liouville theorems involving the arithmetic polynomial divisor functions. We then use the results obtained to establish several polynomials identities over the finite fields of elements. As an example, we show that some complex sums of polynomials in may be reduced to very simple algebraic expressions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
An earlier version of this paper was presented in 12th Annual International Conference on Mathematics and Statistics: Teaching, Theory and Applications, which took place in July 2018 in Athens, Greece. The author would like to sincerely thank and encourage the three undergraduate students (Tasnim I. Kreishan, Nada M. Alsereidi, and Fatima A. Albreiki) for their involvement in the project during the summer and fall 2017. The author is also very grateful to the Research and Sponsored Office at the United Arab Emirates University for accepting to fund this project. This work was supported by the Summer Undergraduate Research Program [SURE PLUS 2017] at the United Arab Emirates University [Grant 31S267 (G00002400)].