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Distribution of Complex Algebraic Numbers on the Unit Circle

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For −π ≤ β1 < β2 ≤ π, denote by Φβ1,β2(Q) the amount of algebraic numbers of degree 2m, elliptic height at most Q, and arguments in [β1, β2], lying on the unit circle. It is proved that

$$ {\Phi}_{\beta_1,{\beta}_2}(Q)={Q}^{m+1}\underset{\beta_1}{\overset{\beta_2}{\int }}p(t) dt+O\left({Q}^m\log Q\right),\kern1em Q\to \infty, $$

where p(t) coincides up to a constant factor with density of the roots of a random trigonometrical polynomial. This density is calculated explicitly using the Edelman–Kostlan formula.

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Author information

Authors and Affiliations

  1. Bielefeld University, Bielefeld, Germany

    F. Götze & A. Gusakova

  2. Münster University, Münster, Germany

    Z. Kabluchko

  3. St.Petersburg Department of Steklov Mathematical Institute, St.Petersburg, Russia

    D. Zaporozhets

Authors
  1. F. Götze
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  2. A. Gusakova
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  3. Z. Kabluchko
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  4. D. Zaporozhets
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Correspondence to F. Götze.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 90–107.

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Götze, F., Gusakova, A., Kabluchko, Z. et al. Distribution of Complex Algebraic Numbers on the Unit Circle. J Math Sci 251, 54–66 (2020). https://doi.org/10.1007/s10958-020-05064-w

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