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Irrationalité de valeurs de zêta

Stéphane Fischler (2002/2003)

Séminaire Bourbaki

Les valeurs aux entiers pairs (strictement positifs) de la fonction ζ de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de π . En revanche, on sait très peu de choses sur la nature arithmétique des ζ ( 2 k + 1 ) , pour k ≥ 1 entier. Apéry a démontré en 1978 que ζ ( 3 ) est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de ζ ( 2 k + 1 ) sont irrationnels, mais sans pouvoir en exhiber aucun autre que ζ ( 3 ) . Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques...

Irrationality of quick convergent series

Jaroslav Hančl (1996)

Journal de théorie des nombres de Bordeaux

We generalize a previous result due to Badea relating to the irrationality of some quick convergent infinite series.

Irreducibility of automorphic Galois representations of G L ( n ) , n at most 5

Frank Calegari, Toby Gee (2013)

Annales de l’institut Fourier

Let π be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GL n ( 𝔸 F ) , where F is a totally real field and n is at most 5 . We show that for all primes l , the l -adic Galois representations associated to π are irreducible, and for all but finitely many primes l , the mod l Galois representations associated to π are also irreducible. We also show that the Lie algebras of the Zariski closures of the l -adic representations are independent of l .

Irreducibility of the iterates of a quadratic polynomial over a field

Mohamed Ayad, Donald L. McQuillan (2000)

Acta Arithmetica

1. Introduction. Let K be a field of characteristic p ≥ 0 and let f(X) be a polynomial of degree at least two with coefficients in K. We set f₁(X) = f(X) and define f r + 1 ( X ) = f ( f r ( X ) ) for all r ≥ 1. Following R. W. K. Odoni [7], we say that f is stable over K if f r ( X ) is irreducible over K for every r ≥ 1. In [6] the same author proved that the polynomial f(X) = X² - X + 1 is stable over ℚ. He wrote in [7] that the proof given there is quite difficult and it would be of interest to have an elementary proof. In the sequel...

Irreducible Sobol' sequences in prime power bases

Henri Faure, Christiane Lemieux (2016)

Acta Arithmetica

Sobol' sequences are a popular family of low-discrepancy sequences, in spite of requiring primitive polynomials instead of irreducible ones in later constructions by Niederreiter and Tezuka. We introduce a generalization of Sobol' sequences that removes this shortcoming and that we believe has the potential of becoming useful for practical applications. Indeed, these sequences preserve two important properties of the original construction proposed by Sobol': their generating matrices are non-singular...

Currently displaying 321 – 340 of 387