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Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

Ilwoo Cho, Palle E. T. Jorgensen (2015)

Special Matrices

In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...

Monogénéité de l'anneau des entiers de certains corps de classes de rayon

Vincent Fleckinger (1988)

Annales de l'institut Fourier

Soient k une extension quadratique imaginaire de Q et A son anneau des entiers. Lorsque 3 est décomposé dans k , nous démontrons que les anneaux d’entiers de certains corps de classe de rayon de k sont monogènes sur l’anneau des entiers du corps de classes de rayon 3. Des générateurs de “monogénéite” sont obtenus a l’aide de fonctions elliptiques qui paramétrisent un modèle de Deuring de la courbe elliptique associée au réseau A .

Multiplicative dependence of shifted algebraic numbers

Paulius Drungilas, Artūras Dubickas (2003)

Colloquium Mathematicae

We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.

Non-Wieferich primes in number fields and a b c -conjecture

Srinivas Kotyada, Subramani Muthukrishnan (2018)

Czechoslovak Mathematical Journal

Let K / ℚ be an algebraic number field of class number one and let 𝒪 K be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in 𝒪 K under the assumption of the a b c -conjecture for number fields.

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Kevin J. McGown (2012)

Journal de Théorie des Nombres de Bordeaux

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant Δ = 7 2 , 9 2 , 13 2 , 19 2 , 31 2 , 37 2 , 43 2 , 61 2 , 67 2 , 103 2 , 109 2 , 127 2 , 157 2 . A large part of the proof is in establishing the following more general result: Let K be a Galois number field of odd prime degree ℓ and conductor f . Assume the GRH for ζ K ( s ) . If 38 ( ℓ - 1 ) 2 ( log f ) 6 log log f < f , then K is not norm-Euclidean.

Currently displaying 121 – 140 of 292