Classical Theory of Arithmetic Functions

Sivaramakrishnan, R.

doi:10.1201/9781315139463

Cited by 41 publications

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“…2.1. Arithmetic functions [1,46,48]. An arithmetic function f maps the positive integers to the complex numbers.…”

Section: Preliminariesmentioning

confidence: 99%

“…Jordan totient functions form an important family of multiplicative arithmetic functions. These functions are a generalization of the Euler totient function [48] and they naturally emerge when computing the value (log Φ n ) (k) (1).…”

Section: Preliminariesmentioning

confidence: 99%

“…If all integer entries ofē are non-negative, then we say that Jē is a Jordan totient product. [48,49]. In this section we recall some relevant properties of cyclotomic polynomials.…”

Section: Preliminariesmentioning

confidence: 99%

“…We have (log Φ n )) (1) (±1) = ±ϕ(n)/2 for n > 2. [10,15,46,48]. Given two positive integers n and k, the Ramanujan sum r k (n) is defined as the (a priori) complex number…”

Section: Cyclotomic Polynomialsmentioning

confidence: 99%

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

Herrera-Poyatos

Moree

2021

Expositiones Mathematicae

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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.

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“…2.1. Arithmetic functions [1,46,48]. An arithmetic function f maps the positive integers to the complex numbers.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…If all integer entries ofē are non-negative, then we say that Jē is a Jordan totient product. [48,49]. In this section we recall some relevant properties of cyclotomic polynomials.…”

Section: Preliminariesmentioning

confidence: 99%

“…We have (log Φ n )) (1) (±1) = ±ϕ(n)/2 for n > 2. [10,15,46,48]. Given two positive integers n and k, the Ramanujan sum r k (n) is defined as the (a priori) complex number…”

Section: Cyclotomic Polynomialsmentioning

confidence: 99%

See 2 more Smart Citations