Eigenvalues in the large sieve inequality, II

Ramaré, Olivier

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 181-196.

Nous explorons numériquement les valeurs de la forme hermitienne

$$\sum _{q\le Q}\sum _{amod{}^{*}q}{\left|\sum _{n\le N}{\phi }_{n}e(na/q)\right|}^2$$

lorsque $N={\sum }_{q\le Q}\phi \left(q\right)$. 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

$$\sum _{q\le Q}\sum _{amod{}^{*}q}{\left|\sum _{n\le N}{\phi }_{n}e(na/q)\right|}^2$$

when $N={\sum }_{q\le Q}\phi \left(q\right)$. 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.

Classification : 11L03, 11L07, 11L26, 30E05, 41A10, 41A17
Mots clés : Large sieve inequality, circle method, Jackson polynomials, Hausdorff moment problem

[1] E. Bombieri, Le grand crible dans la théorie analytique des nombres. Astérisque 18 (1987). | Numdam | MR | Zbl

[2] J.M. Borwein and A.S. Lewis, On the convergence of moment problems. Trans. Amer. Math. Soc. 325(1) (1991), 249–271. | MR | Zbl

[3] G. Greaves, An algorithm for the Hausdorff moment problem. Numerische Mathematik 39(2) (1982), 231–238. | EuDML | MR | Zbl

[4] F. Hausdorff, Summationsmethoden und Momentfolgen. I. Math. Z. 9 (1921), 74–109. | EuDML | JFM | MR

[5] F. Hausdorff, Summationsmethoden und Momentfolgen. II. Math. Z. 9 (1921), 280–299. | EuDML | JFM | MR

[6] F. Hausdorff, Momentenprobleme für ein endliches Intervall. Math. Z. 16 (1923), 220–248. | EuDML | JFM | MR

[7] H.L. Montgomery, Topics in Multiplicative Number Theory. Lecture Notes in Mathematics (Berlin) 227. Springer–Verlag, Berlin–New York, 1971. | MR | Zbl

[8] I.J. Schoenberg, R. Askey and A. Sharma, Hausdorff’s moment problem and expansions in Legendre polynomials. J. Math. Anal. Appl. 86 (1982), 237–245. | MR | Zbl

[9] O. Ramaré, Eigenvalues in the large sieve inequality. Funct. Approximatio, Comment. Math. 37 (2007), 7–35. | MR | Zbl

[10] A. Selberg, Collected papers. Springer–Verlag, Berlin, 1991. | MR | Zbl

[11] J. Szabados, On the convergence and saturation problem of the Jackson polynomials. Acta Math. Acad. Sci. Hungar. 24 (1973), 399–406. | MR | Zbl