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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 31, 2015 12:30, Moscow, Steklov Mathematical Institute of the Russian Academy of Sciences

On the asymptotic distribution of algebraic numbers on the real axis

D. V. Koleda

Institute of Mathematics of the National Academy of Sciences of Belarus

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 in [1],[2]. There, in fact, the density function of real algebraic numbers was correctly defined and explicitly described. In the talk, we will discuss the results [1],[2] on the distribution of real algebraic numbers.