# Distribution of Complex Algebraic Numbers on the Unit Circle

@article{Gotze2018DistributionOC, title={Distribution of Complex Algebraic Numbers on the Unit Circle}, author={Friedrich Gotze and Anna Gusakova and Zakhar Kabluchko and Dmitry Zaporozhets}, journal={arXiv: Number Theory}, year={2018} }

For $-\pi\leq\beta_1<\beta_2\leq\pi$ denote by $\Phi_{\beta_1,\beta_2}(Q)$ the number of algebraic numbers on the unit circle with arguments in $[\beta_1,\beta_2]$ of degree $2m$ and with elliptic height at most $Q$. We show that \[ \Phi_{\beta_1,\beta_2}(Q)=Q^{m+1}\int\limits_{\beta_1}^{\beta_2}{p(t)}\,{\rm d}t+O\left(Q^m\,\log Q\right),\quad Q\to\infty, \] where $p(t)$ coincides up to a constant factor with the density of the roots of some random trigonometric polynomial. This density is… Expand

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