Distribution of complex algebraic numbers

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.

No reviews yet.

No comments yet.