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 Chebyshevskii Sb., 2015, Volume 16, Issue 1, Pages 191–204 (Mi cheb375)

INTERNATIONAL CONFERENCE IN MEMORY OF A. A. KARATSUBA ON NUMBER THEORY AND APPLICATIONS

On the asymptotic distribution of algebraic numbers with growing naive height

D. V. Koleda

Institute of Mathematics of the National Academy of Sciences of Belarus

Abstract: Till recently, even for quadratic algebraic numbers, it was unknown, how frequently do algebraic numbers appear in an arbitrary interval depending on its position and length.
Let $\mathbb{A}_n$ be the set of algebraic numbers of $n$-th degree, and let $H(\alpha)$ be the naive height of $\alpha$ that equals to the naive height of its minimal polynomial by definition. The above problem comes to the study of the following function:
$$\Phi_n(Q, x) := # \{ \alpha \in \mathbb{A}_n \cap \mathbb{R} : H(\alpha)\le Q, \alpha < x \}.$$
The exact asymptotics of $\Phi_n(Q,x)$ as $Q\to +\infty$ was recently obtained by the author. There, in fact, the density function of real algebraic numbers was correctly defined and explicitly described. In the paper, we discuss the results on the distribution of real algebraic numbers. For $n=2$, we improve an estimate of a remainder term in the asymptotics of $\Phi_2(Q,x)$, and obtain the following formula:
$$\Phi_2(Q, +\infty) = \lambda Q^3 - \kappa Q^2 \ln Q + O(Q^2),$$
where $\lambda$ and $\kappa$ are effective constants.
Bibliography: 16 titles.

Keywords: algebraic numbers, generalized Farey series, integral polynomials.

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UDC: 511.35, 511.48, 511.75

Citation: D. V. Koleda, “On the asymptotic distribution of algebraic numbers with growing naive height”, Chebyshevskii Sb., 16:1 (2015), 191–204

Citation in format AMSBIB
\Bibitem{Kol15} \by D.~V.~Koleda \paper On the asymptotic distribution of algebraic numbers with growing naive height \jour Chebyshevskii Sb. \yr 2015 \vol 16 \issue 1 \pages 191--204 \mathnet{http://mi.mathnet.ru/cheb375} \elib{http://elibrary.ru/item.asp?id=23384584} 

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This publication is cited in the following articles:
1. E. A. Karatsuba, M. A. Korolev, I. S. Rezvyakova, V. N. Chubarikov, “O konferentsii pamyati Anatoliya Alekseevicha Karatsuby po teorii chisel i prilozheniyam”, Chebyshevskii sb., 16:1 (2015), 89–152
2. F. Götze, D. Kaliada, D. N. Zaporozhets, “Correlations between real conjugate algebraic numbers”, Chebyshevskii sb., 16:4 (2015), 90–99
3. D. V. Koleda, “O tselykh algebraicheskikh chislakh i unitarnykh mnogochlenakh vtoroi stepeni”, Chebyshevskii sb., 17:1 (2016), 117–129
4. D. Koleda, “On a way of ordering real algebraic numbers uniformly”, Proc. Steklov Inst. Math., 296, suppl. 2 (2017), 61–69
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