[Paper Review] Values of coefficients of cyclotomic polynomials II

Chun-Gang Ji, Wei-Ping Li+1 more|arXiv (Cornell University)|Nov 30, 2007

Analytic Number Theory Research1 references4 citations

TL;DR

This paper proves that for any positive integer $ m $, the coefficients of the cyclotomic polynomials $ \Phi_{mn}(x) $ and their reciprocals $ 1/\Phi_{mn}(x) $, as $ n $ ranges over positive integers, attain every integer value. The proof uses properties of Möbius inversion, Dirichlet’s theorem on primes in arithmetic progressions, and modular expansions of generating functions to show that coefficient sets $ S(m) $ and $ R(m) $ are equal to $ \mathbb{Z} $, generalizing prior results for prime powers to all integers $ m $.

ABSTRACT

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. The first two authors showed in part I that if m is a prime power and n and k range over the non-negative integers, then a(mn,k) assumes every integer value. Here this result is extended to the case where m is arbitrary. The proof use some properties of reciprocal cyclotomic polynomials (see arXiv:0709.1570).

Motivation & Objective

• To extend the result that coefficient sets of cyclotomic polynomials cover all integers from the case of prime powers to arbitrary positive integers.
• To establish that the set of coefficients $ a(mn,k) $ and $ c(mn,k) $ of $ \Phi_{mn}(x) $ and $ 1/\Phi_{mn}(x) $, respectively, equals $ \mathbb{Z} $ for any fixed $ m $.
• To unify and generalize earlier results by Suzuki (for $ m=1 $), the first two authors (for $ m = p^e $), and Moree (for reciprocal coefficients).
• To provide a uniform proof strategy using modular expansions and prime distribution in arithmetic progressions, avoiding case distinctions present in earlier works.

Proposed method

• Use Möbius inversion to express $ \Phi_n(x) $ as a product over $ (1 - x^d)^{\mu(n/d)} $, enabling modular analysis.
• Apply Lemma 1 to show that $ c(n,k) $ depends only on $ k \mod n $, ensuring periodicity of reciprocal coefficients.
• Use Lemma 2 and Corollary 1 to reduce the problem to squarefree $ m $, since $ S(m) = S(\kappa(m)) $ and $ R(m) = R(\kappa(m)) $.
• Leverage quantitative Dirichlet’s theorem (Lemma 3) to find $ t $ primes $ p_j \equiv 1 \mod m $ in the interval $ (n, 15n/8) $, ensuring sufficient density.
• Construct $ m_1 $ as a product of such primes (with a possible extra prime $ q > 2p_1 $) so that $ \mu(m_1) = -1 $, enabling controlled expansion of $ \Phi_{m_1 m}(x) \mod x^{2p_1} $.
• Derive a recurrence for $ a(m_1 m, k) $ in terms of $ c(m,k) $ and $ c(m,k-p_j) $, using the identity $ a(m_1 m, k) = c(m,k) - \mu(m)t c(m,k-1) $ for $ p_t \leq k < 2p_1 $, and analyze cases based on $ \mu(m) = \pm 1 $.

Experimental results

Research questions

• RQ1 Can the coefficient set $ S(m) = \{ a(mn,k) \mid n \geq 1, k \geq 0 \} $ cover all integers $ \mathbb{Z} $ for any fixed positive integer $ m $, not just prime powers?
• RQ2 Does the reciprocal coefficient set $ R(m) = \{ c(mn,k) \mid n \geq 1, k \geq 0 \} $ also equal $ \mathbb{Z} $ for arbitrary $ m $, extending known results for $ m=1 $ and $ m=p^e $?
• RQ3 Can the proof strategy be unified to avoid case distinctions based on parity or number of prime factors of $ m $, as in prior works?
• RQ4 To what extent can the density of primes in arithmetic progressions be used to control coefficient growth and ensure all integers are attained?

Key findings

• For any fixed $ m \geq 1 $, the set of coefficients $ a(mn,k) $ of $ \Phi_{mn}(x) $ covers all integers: $ S(m) = \mathbb{Z} $, generalizing the prior result for $ m = p^e $.
• Similarly, the set of coefficients $ c(mn,k) $ of the reciprocal series $ 1/\Phi_{mn}(x) $ also covers all integers: $ R(m) = \mathbb{Z} $.
• The proof shows that for $ \mu(m) = 1 $, coefficients $ a(m_1 m, p_t + 1) = 1 - t $ and $ a(m_1 m, p_t + q_2) = t - 1 $, which cover all integers as $ t $ varies.
• For $ \mu(m) = -1 $, coefficients $ a(m_1 m, p_t) = -1 + t $ and $ a(m_1 m, p_t + 1) = -t $ or $ 1 - t $, again covering $ \mathbb{Z} $ as $ t \to \infty $.
• The use of $ x^{2p_1} $-modular expansion simplifies the proof compared to earlier works, eliminating the need for case distinctions based on parity of $ m $.
• The constant $ 15/8 $ in the interval length can be replaced by $ 2 - \epsilon $ for any $ \epsilon > 0 $, with corresponding adjustments to the lower bound on $ n $.