On the periods of the linear congruential and power generators
On the splitting of primes in an arithmetic progression. II
Orders of quadratic extensions of number fields
Paramétrisation de structures algébriques et densité de discriminants
La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant , sous l’action de par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant . Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers,...
Piatetski-Shapiro meets Chebotarev
Let K be a finite Galois extension of the field ℚ of rational numbers. We prove an asymptotic formula for the number of Piatetski-Shapiro primes not exceeding a given quantity for which the associated Frobenius class of automorphisms coincides with any given conjugacy class in the Galois group of K/ℚ. In particular, this shows that there are infinitely many Piatetski-Shapiro primes of the form a² + nb² for any given natural number n.
Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields
Let be a linear integer recurrent sequence of order , and define as the set of primes that divide at least one term of . We give a heuristic approach to the problem whether has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that has positive lower density for “generic” sequences . Some numerical examples are included.
Prime divisors of Lucas sequences
Quadratic fields with infinite Hilbert 2-class field towers
Quelques applications du théorème de densité de Chebotarev
Reductions of an elliptic curve with almost prime orders
Some problems in multidimensional analytic number theory
Steinitz classes and discriminant counting
Steinitz classes of central simple algebras
Sur une version algébrique de la notion de sous-groupe minimal relatif de
The Cebotarev Density Theorem for Function Fields: An Elementary Approach.
The correction factor in Artin's primitive root conjecture
In 1927, E. Artin proposed a conjectural density for the set of primes for which a given integer is a primitive root modulo . After computer calculations in 1957 by D. H. and E. Lehmer showed unexpected deviations, Artin introduced a correction factor to explain these discrepancies. The modified conjecture was proved by Hooley in 1967 under assumption of the generalized Riemann hypothesis. This paper discusses two recent developments with respect to the correction factor. The first is of historical...
The Divisibility of Normal Integral Generators.
The least prime ideal.
The number of different lengths of irreducible factorization of a natural number in an algebraic number field
The Piltz divisor problem in number fields: An improved lower bound by Soundararajan's method