ON ALGEBRAIC INTEGERS AND MONIC POLYNOMIALS OF SECOND DEGREE
https://doi.org/10.22405/2226-8383-2016-17-1-117-129
Abstract
In this paper we consider the algebraic integers of second degree and reducible quadratic monic polynomials with integer coefficients. Let Q > 4 be an integer. Define Ωn(Q, S) to be the number of algebraic integers of degree n and height 6 Q belonging to S ⊆ R. We improve the remainder term of the asymptotic formula for Ω2(Q, I), where I is an arbitrary interval. Denote by R(Q) the set of reducible monic polynomials of second degree with integer coefficients and height 6 Q. We obtain the formula #R(Q) = 2 XQ k=1 τ (k) + 2Q + hp Q i − 1, where τ (k) is the number of divisors of k. Besides we show that the number of real algebraic integers of second degree and height 6 Q has the asymptotics Ω2(Q,R) = 8Q2 −16 3 Q p Q − 4QlnQ + 8(1 − γ)Q + O p Q , where γ 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.
About the Author
D. V. Koleda
Institute of Mathematics of the National Academy of Sciences of Belarus
Russian Federation
junior researcher
Russian Federation
junior researcher
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Review
For citations:
Koleda D.V. ON ALGEBRAIC INTEGERS AND MONIC POLYNOMIALS OF SECOND DEGREE. Chebyshevskii Sbornik. 2016;17(1):117-129. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-1-117-129
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