Infinite products involving Dirichlet characters and cyclotomic polynomials

Infinite Products of Cyclotomic Polynomials

We study analytic properties of certain infinite products of cyclotomic polynomials that generalize some introduced by Mahler. We characterize those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behavior near roots of unity.

Composed products and factors of cyclotomic polynomials over finite fields

Let q = ps be a power of a prime number p and let Fq be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over Fq . In particular we obtain the explicit factorization of the cyclotomic polynomial Φ...

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Cyclotomic Polynomials

The cyclotomic polynomials

1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(1 2) = −1 and e(s + t) = e(s)e(t) for all s, t. Consider the polynomial x n − 1. The complex factorisation is obvious: the zeros of the polynomial are e(k/n) for 1 ≤ k ≤ n, so x n − 1 = n k=1 x − e k n. (1) One of these factors is x − 1, and when n is even, another is x + 1. The remaini...