Distribution of complex algebraic numbers

Abstract

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

11 pages; beginning of introduction shortened; mistypes corrected

References in corpus (3)

• On the density function of the distribution of real algebraic numbers
• Correlations between real conjugate algebraic numbers
• Discriminant and root separation of integral polynomials