On algebraic integers and monic polynomials of second degree

D. V. Koleda

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 117-129

Résumé

In this paper we consider the algebraic integers of second degree and reducible quadratic monic polynomials with integer coefficients. Let $Q\ge 4$ be an integer. Define $\Omega_n(Q,S)$ to be the number of algebraic integers of degree $n$ and height $\le Q$ belonging to $S\subseteq\mathbb{R}$. We improve the remainder term of the asymptotic formula for $\Omega_2(Q,I)$, where $I$ is an arbitrary interval. Denote by $\mathcal{R}(Q)$ the set of reducible monic polynomials of second degree with integer coefficients and height $\le Q$. We obtain the formula $$ \#\mathcal{R}(Q) = 2 \sum_{k=1}^Q \tau(k) + 2Q + \left[\sqrt{Q}\right] - 1, $$ where $\tau(k)$ is the number of divisors of $k$. Besides we show that the number of real algebraic integers of second degree and height $\le Q$ has the asymptotics $$ \Omega_2(Q,\mathbb{R}) = 8 Q^2 - \frac{16}{3}Q\sqrt{Q} - 4Q\ln Q + 8(1-\gamma) Q + O\!\left(\sqrt{Q}\right), $$ where $\gamma$ is the Euler constant. It is known that the density function of the distribution of algebraic integers of degree $n$ uniformly tends to the density function of algebraic numbers of degree $n-1$. We show that for $n=2$ the integral of their difference over the real line has nonzero limit as height of numbers tends to infinity. Bibliography: 17 titles.

Détail

Keywords: algebraic integers, distribution of algebraic integers, quadratic irrationalities, integral monic polynomials.