Eigenvalues in the large sieve inequality, II

Olivier Ramaré[1]

  • [1] Laboratoire CNRS Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq cedex, France

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 1, page 181-196
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Ramaré, Olivier. "Eigenvalues in the large sieve inequality, II." Journal de Théorie des Nombres de Bordeaux 22.1 (2010): 181-196. <http://eudml.org/doc/116394>.

@article{Ramaré2010,
abstract = {We explore numerically the eigenvalues of the hermitian form\begin\{equation*\} \sum \_\{q\le Q\}\sum \_\{a~\@mod \;^* q\}\Bigl |\sum \_\{n\le N\}\varphi \_n e(na/q)\Bigr |^2 \end\{equation*\}when $N=\sum _\{q\le Q\}\phi (q)$. 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.},
affiliation = {Laboratoire CNRS Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq cedex, France},
author = {Ramaré, Olivier},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Large sieve inequality; circle method; Jackson polynomials; Hausdorff moment problem; large sieve; eigenvalues; upper bound; numerical study; limiting distribution},
language = {eng},
number = {1},
pages = {181-196},
publisher = {Université Bordeaux 1},
title = {Eigenvalues in the large sieve inequality, II},
url = {http://eudml.org/doc/116394},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Ramaré, Olivier
TI - Eigenvalues in the large sieve inequality, II
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 181
EP - 196
AB - We explore numerically the eigenvalues of the hermitian form\begin{equation*} \sum _{q\le Q}\sum _{a~\@mod \;^* q}\Bigl |\sum _{n\le N}\varphi _n e(na/q)\Bigr |^2 \end{equation*}when $N=\sum _{q\le Q}\phi (q)$. 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.
LA - eng
KW - Large sieve inequality; circle method; Jackson polynomials; Hausdorff moment problem; large sieve; eigenvalues; upper bound; numerical study; limiting distribution
UR - http://eudml.org/doc/116394
ER -

References

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  7. H.L. Montgomery, Topics in Multiplicative Number Theory. Lecture Notes in Mathematics (Berlin) 227. Springer–Verlag, Berlin–New York, 1971. Zbl0216.03501MR337847
  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. Zbl0483.44012MR649868
  9. O. Ramaré, Eigenvalues in the large sieve inequality. Funct. Approximatio, Comment. Math. 37 (2007), 7–35. Zbl1181.11059MR2363835
  10. A. Selberg, Collected papers. Springer–Verlag, Berlin, 1991. Zbl0729.11001MR1295844
  11. J. Szabados, On the convergence and saturation problem of the Jackson polynomials. Acta Math. Acad. Sci. Hungar. 24 (1973), 399–406. Zbl0269.42003MR346399

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