On the distribution of Salem numbers

JOURNAL OF NUMBER THEORY

卷:216

Article

Gotze, Friedrich¹  Gusakova, Anna^1,2 

[1] Bielefeld Univ, Univ Str 25, D-33615 Bielefeld, Germany

[2] Ruhr Univ Bochum, Univ Str 150, D-44801 Bochum, Germany

关键词: Salem number;    Counting formula;    Correlation function;    Jacobi random matrix ensemble;    Weil height;   

DOI  :  10.1016/j.jnt.2020.02.012

来源: Elsevier

【 摘 要 】

In- this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals I-1, ..., I-k subset of [0; pi], 1 <= k <= m let Sal(m, k)(Q, I-1, ..., I-k) denote the set of ordered (k + 1)-tuples (alpha(0), ..., alpha(k)) of conjugate algebraic integers, such that alpha(0 )is a Salem number of degree 2m + 2 satisfying alpha(0) <= Q for some positive real number Q and arg alpha(i) is an element of I-i We derive the following asymptotic approximation #Sal(m,k) (Q, I-1, ..., I-k) =omega(m) Q(m+1) integral(I1 )... integral(Ik )rho(m,k)(theta)d theta +O(Q(m)), Q ->infinity, providing explicit expressions for the constant omega(m) and the function rho(m,k) (theta). Moreover we derive a similar asymptotic formula for the set of all Salem numbers of fixed degree and absolute value bounded by Q as Q ->infinity. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free