Coefficients and higher order derivatives of cyclotomic polynomials: Old and new

Abstract

The n th cyclotomic polynomial Φn(x) is the minimal polynomial of an n th primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers n. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach to derive some of them, leading also to shorter proofs as a by-product. In particular, we show that some of the formulas have a more elegant reinterpretation in terms of Bell polynomials. This approach amounts to computing the logarithmic derivatives of Φn at certain points. Furthermore, we show that the logarithmic derivatives at ±1 of any Kronecker polynomial (a monic product of cyclotomic polynomials and a monomial) satisfy a family of linear equations whose coefficients are Stirling numbers of the second kind. We apply these equations to show that certain polynomials are not Kronecker. In particular, we infer that for every k ≥ 4 there exists a symmetric numerical semigroup with embedding dimension k and Frobenius number 2k + 1 that is not cyclotomic, thus establishing a conjecture of Alexandru Ciolan, Pedro García-Sánchez and the second author. In an appendix Pedro García-Sánchez shows that for every k ≥ 4 there exists a symmetric non-cyclotomic numerical semigroup having Frobenius number 2k + 1. 1 arXiv:1805.05207v2 [math.NT] 22 Aug 2018Proof. The result follows on invoking (5.5) and noting that for every positive integer j we haved|n J j (d) − J j (n) = (2 j − 1)(n j − J j (n)).The following corollary is a consequence of the previous identity and its proof is similar to that of Theorem 5.8.

Jordan totient quotients

“…with χ a Dirichlet character of modulus m. We will see such a result for −1 in case k = 2 in the proof of Theorem 4, which is due to Herrera-Poyatos and the first author [4]. Finally, in Theorem 5, we determine the average of the Schwarzian derivative of Φ n (z) evaluated at z = 1.…”

“…Proof. By [4,Corollary 22] it follows that for n ≥ 3 we have Φ ′′ n (−1) Φ n (−1) = ϕ(n) 4 (ϕ(n) + a n Ψ(n) − 2) , where a n =      1 if n is odd, 1/9 if 2 n, 1/3 otherwise.…”

“…a survey (and some new results) see Herrera-Poyatos and Moree[4]. The Bernoulli numbers B n can be recursively defined byB n = − n − k + 1 , with B 0 = 1.…”

On Symmetric But Not Cyclotomic Numerical Semigroups

“…Therefore CY e ‰ SY e . This conjecture was proved independently by Herrera-Poyatos and Moree in [4] by different methods.…”

“…This conjecture was also proved independently by different methods by Herrera-Poyatos and Moree. [4] Definition. For every nonnegative integer t and positive integer n ě 6t`2, let S n,t be the numerical semigroup generated by:…”