On the distribution of numbers related to the divisors of $x^n-1$

TL;DR

This paper investigates the distribution of certain divisors of cyclotomic polynomials, establishing that the set of integers with specified divisor coefficients has a well-defined natural density and calculating its value.

Contribution

It proves the existence of the natural density for the set of integers related to divisor coefficients of $x^n-1$ and explicitly computes this density.

Findings

The set of integers with specified divisor coefficients has a natural density.

The exact value of this natural density is determined.

Abstract

Let $n_1,\cdots,n_r$ be any finite sequence of integers and let $S$ be the set of all natural numbers $n$ for which there exists a divisor $d(x)=1+\sum_{i=1}^{deg(d)}c_ix^i$ of $x^n-1$ such that $c_i=n_i$ for $1\leq i \leq r$. In this paper we show that the set $S$ has a natural density. Furthermore, we find the value of the natural density of $S$.