Upper Bounds for Cyclotomic Numbers

Proceedings of the American Mathematical Society, 1975

The cyclotomic numbers of order seven are given in terms of the solutions of a certain system of three quadratic diophantine equations. This is analogous to L. E. Dickson’s evaluation of the cyclotomic numbers of order five, and is a convenient approach for applications to the theory of power residues.

Manuscripta Mathematica, 1991

Let p ≥ 5 and q = pf + 1 be prime numbers, and let s be a primitive root mod q. For 1 ≤ n ≤ p − 2, denote by Jn the Jacobi sum − q−1 k=2 ζ inds(k)+n inds(1−k) p. We study the integers d n,k such that Jn = p−1 k=0 d n,k ζ k p and p−1 k=0 d n,k = 1. We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of Z[ζp + ζ −1 p ], in particular to the study of Vandiver's conjecture. For m ∈ Z − qZ, let ρn(m) be the number of distinct roots of X n+1 − X n + m in Z/qZ. We show that d n,k = f − f −1 a=0 ρn(s k+pa). We give closed formulas for the numbers d 1,k and d 2,k in terms of quadratic and cubic power residue symbols mod q.

Glasgow Mathematical Journal, 1985

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler's function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m

arXiv (Cornell University), 2024

The Brauer-Siegel theorem concerns the size of the product of the class number and the regulator of a number field K. We derive bounds for this product in case K is a prime cyclotomic field, distinguishing between whether there is a Siegel zero or not. In particular, we make a result of Tatuzawa (1953) more explicit. Our theoretical advancements are complemented by numerical illustrations that are consistent with our findings.

Compositio Mathematica

L'accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 223 Some remarks on conjectures about cyclotomic fields and Kgroups of Z

Journal of the Australian Mathematical Society, 2015

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called${\rm\Phi}_{n}^{\ast }(q)$, which is closely related to the cyclotomic polynomial${\rm\Phi}_{n}(x)$and to primitive prime divisors of$q^{n}-1$. Our definition of${\rm\Phi}_{n}^{\ast }(q)$is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants$c$and$k$, we provide an algorithm for determining all pairs$(n,q)$with${\rm\Phi}_{n}^{\ast }(q)\leq cn^{k}$. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

arXiv: Number Theory, 2019

Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being $r(5231)=1.556562\dotsc$). The program used and the results described here, are collected at the following address \url{this http URL}. This is a (preliminary) report about the computational part of a joint project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.

Acta Arithmetica

arXiv (Cornell University), 2024

Kummer's conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer's ratio under three scenarios: no Siegel zero, presence of Siegel zero and assuming the Riemann Hypothesis for the Dirichlet L-series attached to odd characters only. The numerical work in this paper extends and improves on our earlier preprint and demonstrates our theoretical results.

ITM Web of Conferences, 2017

We investigate three classes of Ding-Helleseth-generalized cyclotomic sequences of length pq. We derive the linear complexity and the minimal polynomial of above-mentioned sequences over the finite fields of orders p and q, where p and q are two odd distinct primes, and obtain series of sequences with high linear complexity.