The Measures Of Totally Positive Algebraic Integer

Let α be a totally positive algebraic integer of degree d,and P(x)be its minimal polynomial P(x)=xd+b1xd-1+…+bd-1x+bd=(?)(x-αi),whose.all conjugates α1=α,α2,…,αd are positive real numbers.The Mahler measure of a is M(α)=(?)max(1,αi),and the absolute Mahler measure is Ω(α)=M(α)1/d.The Length of α is L(α)=(?)|bi|+1,and the absolute Length is(?)(α)=L(α)1/d.The R2 measure of α is R2(α)=(?)(1+αi2)1/2,and the absolute R2 measure is r2(α)=R2(α)1/d.The Zhang-Zagier measure of a is Z(α)=M(α)M(1-α),and the absolute Zhang-Zagier measure is ζ(α)=Z(α)1/d.In this paper,we mainly discuss the lower bound of the absolute measures of totally positive algebraic integer and the relationship between different absolute measures.We prove that all but finitely many positive algebraic integers α hasΩ(α)≥1.722667,r2((α)≥1.867859,ζ(α)≥2.359922.In addition,we discuss the lower bound of the absolute Mahler measure of 1-α,and prove that Ω(1-α)=X(α)≥1.358927.At the same time,we show that all but finitely many positive algebraic integersα hasΩ(α)≤0.629202 r2(α)1.613077,1.360105 Ω(α)≤ζ(α)≤0.277763 Ω(α)4.111836,0.986315(?)(α)≤ζ(α)≤0.018699 L(α)5.732541,1.250162 r2(α)≤ζ(α)≤0.114124 r2(α)4.976957.