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 Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 117–129 (Mi cheb457)

On algebraic integers and monic polynomials of second degree

D. V. Koleda

Institute of Mathematics of the National Academy of Sciences of Belarus

Abstract: 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
$$#\mathcal{R}(Q) = 2 \sum_{k=1}^Q \tau(k) + 2Q + [\sqrt{Q}] - 1,$$
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
$$\Omega_2(Q,\mathbb{R}) = 8 Q^2 - \frac{16}{3}Q\sqrt{Q} - 4Q\ln Q + 8(1-\gamma) Q + O(\sqrt{Q}),$$
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.

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

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Document Type: Article
UDC: 511.35, 511.48, 511.75
Accepted:11.03.2016

Citation: D. V. Koleda, “On algebraic integers and monic polynomials of second degree”, Chebyshevskii Sb., 17:1 (2016), 117–129

Citation in format AMSBIB
\Bibitem{Kol16} \by D.~V.~Koleda \paper On algebraic integers and monic polynomials of second degree \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 1 \pages 117--129 \mathnet{http://mi.mathnet.ru/cheb457} \elib{http://elibrary.ru/item.asp?id=25795074}