Eigenvalues in the large sieve inequality, II
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 181-196.

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.

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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.710
Classification : 11L03,  11L07,  11L26,  30E05,  41A10,  41A17
Mots clés : Large sieve inequality, circle method, Jackson polynomials, Hausdorff moment problem
@article{JTNB_2010__22_1_181_0,
     author = {Olivier Ramar\'e},
     title = {Eigenvalues in the large sieve inequality, {II}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {181--196},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.710},
     zbl = {1220.11117},
     mrnumber = {2675879},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.710/}
}
Olivier Ramaré. Eigenvalues in the large sieve inequality, II. Journal de Théorie des Nombres de Bordeaux, Tome 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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