[Paper Review] Reciprocal cyclotomic polynomials

Pieter Moree|arXiv (Cornell University)|Sep 11, 2007

Analytic Number Theory Research11 references7 citations

TL;DR

This paper investigates the coefficients of reciprocal cyclotomic polynomials $Ψ_n(x)$, defined as $(x^n - 1)/\Phi_n(x)$, which are monic integer polynomials whose roots are the non-primitive $n$th roots of unity. It establishes that these coefficients are symmetric in absolute value and often small (flat), but can grow arbitrarily large; notably, the first non-flat case occurs at $n=561$, the smallest Carmichael number, where $c_{561}(17) = -2$. The paper proves that all integers appear as coefficients of $\Psi_{pqr}(x)$ for distinct primes $p<q<r$, demonstrating unbounded coefficient growth.

ABSTRACT

Let $Ψ_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $Ψ_n(x)=(x^n-1)/Φ_n(x)$, with $Φ_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $Ψ_n(x)$ are integers that like the coefficients of $Φ_n(x)$ tend to be surprisingly small in absolute value, e.g. for $n&lt;561$ all coefficients of $Ψ_n(x)$ are $\le 1$ in absolute value. We establish various properties of the coefficients of $Ψ_n(x)$.

Motivation & Objective

• To analyze the structure and coefficient behavior of reciprocal cyclotomic polynomials $\Psi_n(x) = (x^n - 1)/\Phi_n(x)$, which are defined as the product of $\Phi_d(x)$ for all proper divisors $d$ of $n$.
• To understand the extent of coefficient cancellation in $\Psi_n(x)$, analogous to the well-known smallness of coefficients in $\Phi_n(x)$, and to determine when such cancellation fails.
• To characterize the set of coefficients $V_n = \{c_n(k)\}$, especially when they include $-1, 0, 1$, and to identify conditions under which coefficients exceed absolute value 1.
• To investigate whether the coefficients of $\Psi_n(x)$ are bounded or unbounded, and to determine the minimal $n$ for which $|c_n(k)| > 1$, particularly in relation to Carmichael numbers.

Proposed method

• The paper uses Möbius inversion to derive the identity $\Phi_n(x) = \prod_{d|n} (x^d - 1)^{\mu(n/d)}$, which leads to the expression $\Psi_n(x) = \prod_{d|n, d<n} \Phi_d(x)$, ensuring integer coefficients.
• It applies properties of the Möbius function $\mu(n)$ and the radical function $\text{rad}(n)$ to derive recursive identities, such as $\Psi_{2n}(x) = (1 - x^n)\Psi_n(-x)$ for odd $n$, and $\Psi_{pn}(x) = \Psi_n(x^p)$ when $p|n$.
• The paper uses the functional equation $\Psi_n(x) = -x^{n - \varphi(n)} \Psi_n(1/x)$ to establish symmetry: if $c_n(k)$ is a coefficient, so is $-c_n(n - \varphi(n) - k)$, implying $V_n$ is symmetric about zero.
• It proves that $\Psi_n(x)$ is flat (all coefficients in $\{-1, 0, 1\}$) when $n$ has at most two distinct odd prime factors, using structural decomposition and coefficient analysis.
• For higher-order $n$, the paper uses Dirichlet's theorem on primes in arithmetic progressions to construct sequences of primes $p<q<r$ such that $\Psi_{pqr}(x)$ achieves arbitrarily large coefficients.
• It computes explicit examples and uses the structure of $\Phi_{pq}(x)$ from known formulas to derive coefficient values, especially for $n = 561$ and Chernick Carmichael numbers.

Experimental results

Research questions

• RQ1 What is the structure of the coefficients of $\Psi_n(x)$, the reciprocal cyclotomic polynomial, and how do they compare to those of $\Phi_n(x)$?
• RQ2 For which $n$ are the coefficients of $\Psi_n(x)$ bounded in absolute value by 1, and what is the smallest $n$ for which $|c_n(k)| > 1$?
• RQ3 Do all integers appear as coefficients of $\Psi_n(x)$ for some $n$, and if so, for which $n$?
• RQ4 What is the behavior of $\Psi_n(x)$ when $n$ is a Carmichael number, and does $h(\Psi_n(x)) > 1$ hold for all such $n$?
• RQ5 Can the coefficient set $\{c_n(k)\}$ be shown to cover all integers, and what is the minimal $n$ for which $|c_n(k)| = m$ for a given $m$?

Key findings

• The first $n$ for which $|c_n(k)| > 1$ is $n = 561 = 3 \cdot 11 \cdot 17$, where $c_{561}(17) = -2$, and this is the smallest Carmichael number.
• For all $n$ with at most two distinct odd prime factors, $\Psi_n(x)$ is flat: all coefficients are in $\{-1, 0, 1\}$, as shown in Lemma 4.
• The set of coefficients $\{c_n(k)\}$ for $n = pqr$ with distinct primes $p < q < r$ covers all integers: $\{c_{pqr}(k)\} = \mathbb{Z}$, as proven in Theorem 8.
• For Chernick Carmichael numbers $C = (6k+1)(12k+1)(18k+1)$, the coefficient $c_C(24k+2) = -2$, so $h(\Psi_C(x)) = 2$, and these polynomials are not flat.
• The minimal $n$ such that $|c_n(k)| = m$ is given in a table: for $m=2$, $n_0=561$; for $m=3$, $n_0=1155$; for $m=10$, $n_0=11305$, and this $n_0$ remains minimal for $m=10$ to $21$.
• The coefficient $c_n(k)$ satisfies a symmetry: $c_n(k) = -c_n(n - \varphi(n) - k)$, so the coefficient set $V_n$ is symmetric about zero, and if $n - \varphi(n)$ is even, then $c_n((n - \varphi(n))/2) = 0$.