Abstract
In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals I1,…,Ik⊂[0;π], 1≤k≤m let Salm,k(Q,I1,…,Ik) denote the set of ordered (k+1)-tuples (α0,…,αk) of conjugate algebraic integers, such that α0 is a Salem number of degree 2m+2 satisfying α≤Q for some positive real number Q and argαi∈Ii. We derive the following asymptotic approximation#Salm,k(Q,I1,…,Ik)=ωmQm+1∫I1…∫Ikρm,k(θ)dθ+O(Qm),Q→∞, providing explicit expressions for the constant ωm and the function ρm,k(θ). 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→∞.
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