Eigenvalues in the large sieve inequality, II
Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 181-196.

We explore numerically the eigenvalues of the hermitian form

qQamod*qnNφne(na/q)2

when N= qQ 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.

Nous explorons numériquement les valeurs de la forme hermitienne

qQamod*qnNφne(na/q)2

lorsque N= qQ φ(q). Nous améliorons la majoration actuelle et exhibons un graphique conjectural de la distribution asymptotique de ses valeurs propres en exploitant des résultats de calculs intensifs. L’une des conséquences est que la distribution asymptotique existe probablement mais n’est pas absolument continue par rapport à la mesure de Lebesgue.

DOI: 10.5802/jtnb.710
Classification: 11L03, 11L07, 11L26, 30E05, 41A10, 41A17
Keywords: Large sieve inequality, circle method, Jackson polynomials, Hausdorff moment problem

Olivier Ramaré 1

1 Laboratoire CNRS Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq cedex, France
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Olivier Ramaré. Eigenvalues in the large sieve inequality, II. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 181-196. doi : 10.5802/jtnb.710. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.710/

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