Distribution of complex algebraic numbers

TL;DR

This paper derives an asymptotic formula for counting complex algebraic numbers within a region, revealing their distribution pattern as the height bound grows large.

Contribution

It provides a new explicit asymptotic formula for the distribution of algebraic numbers of bounded degree and height in the complex plane.

Findings

Asymptotic count of algebraic numbers in regions for large height bounds

Explicit formula for the density function ta(z)

Error term of order Q^n in the asymptotic estimate

Abstract

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