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Eigenvalues in the large sieve inequality, II

Olivier Ramaré — 2010

Journal de Théorie des Nombres de Bordeaux

We explore numerically the eigenvalues of the hermitian form q Q a mod * q n N ϕ n e ( n a / q ) 2 when N = q Q φ ( q ) . We improve on the existing upper bound, and produce a (conjectural) plot of the asymptotic distribution of its eigenvalues by exploiting fairly extensive computations. The main outcome is that this asymptotic density most probably exists but is not continuous with respect to the Lebesgue measure.

Nombres de racines d’un polynôme entier modulo q

Monique BrantonOlivier Ramaré — 1998

Journal de théorie des nombres de Bordeaux

Nous montrons que l’ensemble des racines modulo une puissance d’un nombre premier d’un polynôme à coefficients entiers de degré d est une union d’au plus d progressions arithmétiques de modules assez grands. Nous en déduisons une majoration du nombre de ses racines dans un intervalle réel court.

Discrepancy estimates for some linear generalized monomials

Roswitha HoferOlivier Ramaré — 2016

Acta Arithmetica

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, ( [ n α ] β ) n 0 , is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show...

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