Correlations between real conjugate algebraic numbers

TL;DR

This paper derives an asymptotic formula for counting conjugate algebraic number tuples within a region, revealing their distribution and correlations as the height bound grows large.

Contribution

It provides an explicit asymptotic count and distribution density for conjugate algebraic numbers of degree up to n, including special cases for n=2.

Findings

Asymptotic formula for the number of conjugate algebraic tuples

Explicit density function for conjugate algebraic numbers

Additional logarithmic factor in the error term when n=2

Abstract

For $B\subset\mathbb{R}^k$ denote by $\Phi_k(Q;B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\leq n$ and naive height $\leq Q$. We show that $$ \Phi_k(Q;B) = \frac{(2Q)^{n+1}}{2\zeta(n+1)} \int_{B} \rho_k(\mathbf{x})\,d\mathbf{x} + O\left(Q^n\right),\quad Q\to \infty, $$ where the function $\rho_k$ will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term.