On the density of irreducible polynomials which generate k-free polynomials over function fields
Introduction
In 1949, Mirsky [6] proved that every sufficiently large number can be written as a sum of a prime and a k-free number for any given . In fact, he derived an asymptotic formula for the number of such representations. In 2001, Yao [9] attempted to show that an analogous result holds in the case of function fields over finite fields. Apparently, according to Math Reviews (MR1841909), there seem to be errors in the paper. This was highlighted as an open problem in Moree's survey [7]. The purpose of this paper is to correct the error in [9] and establish the analogue in the function field case.
Let be a finite field of order q, and let be a fixed polynomial. For , a polynomial is k-free if it is not divisible by the kth power of any irreducible polynomial. In this note, we want to compute the asymptotic density of irreducible polynomials P for which is k-free. For the most part, we adhere to the notation and arrangement of [9]. More precisely, we will prove the following:
Theorem 1 Let be a fixed polynomial, and let be an integer. Denote by the set of all monic irreducible polynomials. We define the set Then we have
Theorem 1 can be directly obtained from the following result and the prime number theorem for :
Theorem 2 Let be a fixed polynomial, and let be an integer. If , then
Section snippets
Preliminaries
We set the following notations throughout this note: For any , we write its factorization with . We define the polynomial Möbius function as Analogously, for any integer , we define It is easy to verify identity
Proof of Theorem 2
We write Since we assumed , using (2.2), we have For any , we can split the sum as We will choose later a suitable t with to minimize the error term (the second term) of (3.2). This is how we fix the mistake in [9], which occurs on lines 10-13, where should
Future directions
As noted earlier, it would be interesting to see if the error term in Theorem 2 can be improved. A generalization of Theorem 1 involving higher dimensional tuples of fixed polynomials can be discussed analogously following the generalization by Mirsky [4] for number fields. Moreover, further generalization is possible following the subsequent result of Mirsky [5].
Acknowledgement
We would like to thank Michael Rosen, Joseph H. Silverman, Wei-Chen Yao, and the anonymous referee for helpful suggestions on an earlier version of this paper.
References (9)
On an elementary density problem for polynomials over finite fields
Finite Fields Appl.
(2001)The arithmetic of polynomials in a Galois field
Am. J. Math.
(1932)- et al.
Additive Number Theory of Polynomials over a Finite Field
(1991) - et al.
Über die Primfunktionen in einer arithmetischen Progression
Math. Z.
(1919)
Cited by (0)
- 1
Research of the first author partially supported by a Coleman Postdoctoral Fellowship.
- 2
Research of the second author partially supported by NSERC Discovery grant.