PUB - Publikationen an der Universität Bielefeld

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.