Jordan totient quotients

Moree, Pieter; Eddin, Sumaia Saad; Sedunova, Alisa; Suzuki, Yoshitake

doi:10.1016/j.jnt.2019.08.014

Cited by 18 publications

(15 citation statements)

References 10 publications

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“…Jordan introduced these generalized totients [14] J l (m) in 1870. The asymptotic behavior of the Jordan totient summatory functions is known:…”

Section: Discussionmentioning

confidence: 99%

Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit

Migdal

2023

Preprint

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This is the second paper in a cycle investigating the exact solution of loop equations in decaying turbulence. We perform numerical simulations of the Euler ensemble, suggested in the previous work, as a solution to the loop equations. We designed novel algorithms for simulation, which take a small amount of computer RAM so that only the CPU time grows linearly with the size of the system and its statistics. This algorithm allows us to simulate systems with billions of discontinuity points on our fractal curve, dual to the decaying turbulence. For the vorticity correlation function, we obtain quantum scaling laws with regime changes between effective indexes drifting with the logarithm of scale. The traditional description of turbulence with fractal or multifractal scaling laws does not apply here. In particular, the effective fractal dimension of our curve is a function of the scale. The measured conditional probabilities of fluctuating variables are smooth functions of the logarithm of scale, with statistical errors being negligible at our large number of random samples $T = 167,772,160$ with $N = 200,000,000$ points on a fractal curve. The quantum jumps arise from the analytical distribution of scaling variable $X = \frac{\cot^2(\pi p/q)}{q^2}$ where $\frac{p}{q}$ is a random fraction. This distribution is related to the totient summatory function and is a discontinuous function of $X$. In particular, the energy spectrum has quantum levels on top of a continuous background.

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“…Jordan introduced these generalized totients [14] J l (m) in 1870. The asymptotic behavior of the Jordan totient summatory functions is known:…”

Section: Discussionmentioning

confidence: 99%

Dual Theory of Decaying Turbulence. 2. Numerical Simulation and Continuum Limit

Migdal

2023

Preprint

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“…This property suggests to call N (k) (z) the normalized k th derivative of f at z and leads to the following problem. Moree et al [39] consider this problem in depth. They take F to be the family of cyclotomic polynomials and z ∈ {−1, 0, 1} and make crucial use of the results in this paper to express the quantities under consideration as linear combinations of Jordan totient quotients.…”

Section: Note Thatmentioning

confidence: 99%

“…exists, as the argument of the sum is a finite Q-linear combination of Jordan totient quotients of non-negative weight, which by [39] are each constant on average. Likewise, several other quantities in this paper, after appropriate normalization, can be shown to be constant on average (actually usually far more precise results than being constant on average can be formulated).…”

Section: Note Thatmentioning

confidence: 99%

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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“…n (1) give important arithmetic functions such as von Mangoldt function and Euler totient function. Lehmer [5] gave an explicit formula of Φ (k) n (1)/Φ n (1) as a polynomial of φ(n) and J 2j (n) over Q, using Stirling numbers and Bernoulli numbers, see [1,2] for further developments. Here we give a quick proof of this fact but without its explicit form.…”

Section: Introductionmentioning

confidence: 99%