## The mean values of totally real algebraic integers.(English)Zbl 0536.12006

Let $$\alpha$$ be a totally real algebraic integer of degree $$d$$ with conjugates $$\alpha_1,\ldots,\alpha_d$$. Let $$p>0$$ and then define $$M_p(\alpha)=(d^{-1}\sum^d_{i=1}| \alpha_i|^d)^{1/p}$$. The author investigates the structure of the set $$\mathcal M_p$$ of values of $$M_p(\alpha)$$ as $$\alpha$$ varies over all totally real algebraic integers of all degrees. For each $$p>0$$, he determines two numbers $$A_p<B_p$$ so that $$\mathcal M_p\cap(1,A_p)$$ is a finite non-empty set whose members are given explicitly, and such that $$\mathcal M_p$$ is dense in $$(B_p,\infty)$$. Thus the smallest element of $$\mathcal M_p$$ is known for all $$p>0$$. The author conjectures that $$B_p$$ is, in fact, the smallest limit point of $$\mathcal M_p$$.
The structure of $$\mathcal M_p$$ is thus similar to the set of values of $$\Omega(\alpha)=\prod \max(1,| \alpha_ i|)^{1/d}$$ which he considered in two earlier papers [J. Aust. Math. Soc., Ser. A 30, 137–149 (1980; Zbl 0457.12001); Math. Comput. 37, 205–208 (1981; Zbl 0475.12001)]. The results improve upon and supersede earlier results of C. L. Siegel $$(p=2)$$ [Ann. Math. (2) 46, 302–312 (1945; Zbl 0063.07009)], J. Hunter $$(p=4)$$ [Proc. Glasg. Math. Assoc. 2, 149–158 (1956; Zbl 0074.03901)] and M. J. McAuley $$(p=1,2,3,4,6,8,10,12)$$ [M. Sc. thesis, Univ. Adelaide, 1981].
The proofs are a nice blend of computation and analysis.

### MSC:

 11R80 Totally real fields 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R04 Algebraic numbers; rings of algebraic integers

### Keywords:

mean values of totally real algebraic integers

### Citations:

Zbl 0063.07009; Zbl 0457.12001; Zbl 0475.12001; Zbl 0074.03901
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