In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals I1,ā¦,Ikā[0;Ļ], 1ā¤kā¤m let Salm,k(Q,I1,ā¦,Ik) denote the set of ordered (k+1)-tuples (Ī±0,ā¦,Ī±k) of conjugate algebraic integers, such that Ī±0 is a Salem number of degree 2m+2 satisfying Ī±ā¤Q for some positive real number Q and argā”Ī±iāIi. We derive the following asymptotic approximation#Salm,k(Q,I1,ā¦,Ik)=ĻmQm+1ā«I1ā¦ā«IkĻm,k(Īø)dĪø+O(Qm),Qāā, providing explicit expressions for the constant Ļm and the function Ļm,k(Īø). Moreover we derive a similar asymptotic formula for the set of all Salem numbers of fixed degree and absolute value bounded by Q as Qāā.
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