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arXiv · 1410.3623

Distribution of complex algebraic numbers

Friedrich Götze·Dzianis Kaliada·Dmitry Zaporozhets
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Abstract

For a region $Ω\subset\mathbb{C}$ denote by $Ψ(Q;Ω)$ the number of complex algebraic numbers in $Ω$ of degree $\leq n$ and naive height $\leq Q$. We show that $$ Ψ(Q;Ω)=\frac{Q^{n+1}}{2ζ(n+1)}\int_Ωψ(z)\,ν(dz)+O\left(Q^n \right),\quad Q\to\infty, $$ where $ν$ is the Lebesgue measure on the complex plane and the function $ψ$ will be given explicitly.

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Friedrich Götze, Dzianis Kaliada, Dmitry Zaporozhets. 2016-03-17. Distribution of complex algebraic numbers. https://arxiv.org/abs/1410.3623

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