The mean values of logarithms of algebraic integers

Dubickas, Artūras

Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 301-313.

Soit ${\alpha }_1=\alpha ,{\alpha }_2,\cdots ,{\alpha }_d$ l’ensemble des conjugués d’un entier algébrique $\alpha$ de degré $d$, n’étant pas une racine de l’unité. Dans cet article on propose de minorer

$$M_p\left(\alpha \right)=\sqrt[p]{\frac{1}{d}{\sum }_{i=1}^d|log|{\alpha }_i{\left|\right|}^p}$$

où $p>1$.

Let $\alpha$ be an algebraic integer of degree $d$ with conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\cdots ,{\alpha }_d$. In the paper we give a lower bound for the mean value

$$M_p\left(\alpha \right)=\sqrt[p]{\frac{1}{d}{\sum }_{i=1}^d|log|{\alpha }_i{\left|\right|}^p}$$

when $\alpha$ is not a root of unity and $p>1$.

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